Nicola Sturgeon has dismissed as "ludicrous" any claims that the surge in Tory support in Scotland could derail her bid to hold a second independence referendum.
The First Minister and SNP admitted the Conservatives in Scotland had "a good day by their standards", with the party returning a record number of councillors north of the border.
But she insisted the SNP had "won this election comfortably" with her party ending up with "more votes, more seats, more councils where we are the largest party, not just compared to every other party but compared to five years ago".
Ms Sturgeon said: "Yes, the Tories made gains and had a good performance by their standards - but that support came from Labour not the SNP, so Labour and the Tories are fighting it out for second place while the SNP continues to be comfortably in first place."
She said: "This was a council election that the SNP fought on local issues, which is probably why the SNP won the election so emphatically.
"But let's take the Tory argument at face value. They chose to fight the election on the issue of an independence referendum, they talked about nothing else, they didn't have any policies for local government.
"So they put that issue centre stage and they lost the election. They came second in the election and the SNP came first.
"If you're going to put a single issue at the centre of your own campaign, then you lose the election, then you're left with a bit of egg on your face and I think the Tories have egg on their face on that question this morning.
"They've had a good day by their standards but we've got to put into context - the Tories polled a lower share of the vote in Scotland than Jeremy Corbyn did in England so, yes, a good performance from a low starting point but it's Labour that the Tories have taken support from, not the SNP."
She spoke out after Scottish Conservatives won a record 276 councillors north of the border - more than double the 115 they secured five years ago - with Tories elected in places such as Paisley's Ferguslie Park, the most deprived part of Scotland
The SNP remains the largest party in local government with 431 councillors voted in, up slightly from its total of 425 in 2012.
But if voting patterns are similar at the General Election on June 8, a surge in Conservative support could see Ruth Davidson's party oust some SNP MPs from Westminster.
Meanwhile, Labour slumped to become the third largest party in Scotland's councils, and was kicked out of power in its Glasgow heartland for the first time in almost 40 years.
Ms Davidson said the results showed "only the Scottish Conservatives have the strength to fight back against the SNP".
She said: "We will speak up for the millions of Scots who have had enough of the uncertainty and division of the last few years.
"We will stand up for everyone who doesn't want a second referendum on independence."
Scottish Labour leader Kezia Dugdale also said there had been a "clear backlash against the SNP's plans for a divisive second independence referendum".
However, Ms Sturgeon said: "I've heard lots of ludicrous arguments in my time in politics but this takes the prize for the most ludicrous argument."
The First Minister also made clear she is confident the SNP will be able to hold on key constituencies the Tories are targeting.
The Conservatives are bidding to oust a number of high-profile nationalists, including Westminster leader Angus Robertson, former first minister Alex Salmond, and Pete Wishart, the chair of the Scottish Affairs Committee, in the General Election.
"We take nothing for granted but I'm confident that the strength of these local candidates will win through," the First Minister stated.
"And as for the General Election, while you've always got to be careful about extrapolating from one election to another, the issue for Scotland becomes quite clearly focused.
"The results in England will tell everybody that the Tories are on course to win the election - the question for Scotland is, do we want to make sure we've got strong voices standing up for Scotland providing opposition to the Tories?
"I think most people will want to see that, which is why it is important to stress the only way to get that is to vote SNP."
She insisted there was "not a shred of disappointment" that the SNP had failed to secure an overall majority in Glasgow City Council, where Labour had been in power since 1980.
Ms Sturgeon said: "I'm over the moon that we are the largest party in Glasgow and are about to form an administration in the city.
"When I first became active in politics in Glasgow the SNP had one councillor in the city chambers and the joke was the Labour vote was weighed, not counted, in Glasgow.
"Now we have every Westminster constituency, every Scottish Parliament constituency and we're about to form an administration.
"The term historic is often overused in politics but to see 40 years of Labour control in Glasgow brought to an end to be replaced by a new SNP city government probably makes that word 'historic' appropriate."
She continued: "This city is bursting with potential and there is really an opportunity now for a bright new start and administration that will see Glasgow fulfil its potential.
"As someone who lives in Glasgow, who represents a Glasgow constituency, I'm really excited now about the opportunities that lie ahead."
Ms Davidson spent the day campaigning in Gordon, where Mr Salmond is the SNP's candidate.
The Scottish Tory leader declared: "This week's local government election has shown we are the only party in Scotland with the strength to fight back against the SNP - in every part of Scotland.
"We won the local government election in Gordon this week, beating the SNP into second place. It means that in this seat, as in many others, it is a two-horse race between us and the Nationalists.
"As is the case right across Scotland, the choice in Gordon is clear.
"It's between a Scottish National Party that will to take us back to more division and instability, and a Scottish Conservative party that will fight against another referendum so we can all move on together."
Scottish Labour deputy leader Alex Rowley meanwhile insisted his party is still a "major party" in local government, despite losing seats.
Mr Rowley stressed: "The SNP failed to win a single majority anywhere in Scotland, and the nature of the voting system means that cross-party deals may be agreed."
A meeting of the party's Scottish Executive Committee - which includes Ms Dugdale, Mr Rowley, trade unions and others - agreed Labour would not do any deals which would result in increased austerity for Scotland's poorest people.
Mr Rowley said: "Labour's approach is clear and consistent: we will categorically refuse to do any deal with another party if it would result in further austerity being imposed on local communities.
"Labour values must run through any deals: the defence of local services against cuts; and the proper funding of the services so many people rely on such as education and care for the elderly.
"Additionally, we will require any power-sharing administrations to protect jobs by opposing any compulsory redundancies.
"Every Labour councillor will always put their local communities first. They will fight against Tory attempts to drive down living standards and will not be distracted by campaigning for a divisive second independence referendum."
Meanwhile James Kelly, the party's general election campaign manager in Scotland, insisted they were looking to win seats from the SNP on June 8.
Labour lost all but one of its Scottish MPs in the 2015 general election, but Mr Kelly said: "Labour was written off before the council elections, but we defied the odds to win three councils, finished joint top in another, and pushed the SNP incredibly close across Scotland.
"As a result we're now targeting a number of gains from the SNP in the General Election in seats where it's clear Labour is the only party that can defeat the nationalists.
"In places such as Glasgow, North Lanarkshire, East Lothian, Inverclyde, Fife and elsewhere, the Tories are in a distant third place. In a first-past-the-post General Election, it's now abundantly clear it is only Labour that can defeat the SNP in these seats."
He added: "If voters want to send Nicola Sturgeon a message to drop her plans for a divisive second independence referendum, they must vote Labour on June 8. A vote for any other party simply risks letting the SNP back in, and increases the risk of another unwanted referendum."
Conservative local government spokesman Graham Simpson MSP said: "The Scottish Conservatives practice devolution as well as preach. So it will be up to our local councillors to decide what is best for their local communities.
"By contrast Labour's proposals suggest that the party's centralising Corbynista wing is taking over.
"It is yet another example of the chaos within Labour - and shows why thousands of Scots have left them to back Ruth Davidson's Scottish Conservatives."