Richard Burgon, a shadow Treasury minister, came under fire for posting a string of tweets putting the party's overall loss of 24 councillors down to previous Labour leaders' records.
The election fought this week under Corbyn marked the fist time since 1985 that an opposition party had lost English council seats.
But Burgon, a new MP and Corbyn ally, came under swift fire from former home secretary Jacqui Smith, who served in top ministerial positions during the Blair and Brown administrations.
She branded Burgon a "disgrace" for his response to Thursday night's results.
In a post on Twitter to the MP, Smith wrote of the "electoral legacy" Corbyn inherited:
Labour MP Rachel Reeves echoed Smith, asking:
To which Burgon issued a short reply:
Criticism of Burgon was echoed by former Conservative MP Louise Mensch too.
"Disastrous legacy of loads of seats put right by Ruth Davidson [the Scottish Tories leader]," she wrote.
But Burgon's analysis did win praise from the 'Jeremy Corbyn for Prime Minister' campaign group, an unofficial team not directly linked to the Labour leader.
Thursday saw Labour pushed into third place in Scotland, behind the SNP and the Conservatives, in what was described as a “painful” night for the party.
Corbyn did defy his harshest critics, as his party's performance was not as bad as some expected, clinging on to councils in the south including Southampton, Crawley and Hastings.
Some predictions had Labour on course to lose as many as 150 local council seats.
In the capital, the party's mayoral candidate Sadiq Khan fended off competition from rival Zac Goldsmith to become London's first Muslim mayor.