The comment, made by the presenter during the programme's Burma special screened in March, was offensive according to the regulator.
Today's ruling by Ofcom involved a scene in which the presenters were involved in the construction of a makeshift bridge over the River Kwai and viewers saw an Asian man walking on the bamboo crossing, while Clarkson declared: "That is a proud moment - but there is a slope on it."
Two viewers were concerned that the word "slope" was an offensive racist term.
An Ofcom spokesperson said: "After a thorough investigation, Ofcom has found the BBC breached broadcasting rules by including an offensive racial term in Top Gear, which was not justified by context.
"Jeremy Clarkson used the word 'slope' to refer both to an Asian man crossing a bridge, and the incline of the bridge. This was scripted in advance. The BBC failed to take the opportunity, either during filming or post-production, to check whether the word had the potential to offend viewers."
The programme's executive producer, Andy Wilman, had previously apologised for the use of the term, saying it had been intended as a "light-hearted word-play joke" and that the team had been unaware the term was considered offensive.
And the BBC said today: "We dealt with this matter some time ago, the programme apologised at the time and explained the context, and we are now focusing on delivering another series of one of Britain's best-loved shows."
The motoring show has faced a number of controversies over the years and prompted complaints from Indian and Mexican politicians over remarks made about their countries while filming on location.
In its response to Ofcom, the BBC said that, although programme-makers "knew that the word could be used to refer to people of Asian origin, they believed that such use was mere slang".
In its judgment, Ofcom said "there was clearly an opportunity both during filming and post-production to research the word and reach a more considered view on whether it was 'mere slang' and had the potential to cause offence to viewers".
The regulator decided there was "insufficient context to justify the broadcast of this material" and that the BBC did not apply generally accepted standards to protect viewers from offensive material.