But Mr Umunna told BBC Radio 4's World at One that "the whole Labour leadership" had to address the failure to capitalise on Government divisions, faltering Brexit negotiations and voter concern over the economy and public services.
"From a Labour point of view there needs to be a proper post-mortem - I think the National Executive Committee should appoint somebody to do that - on this result," said the Streatham MP.
"We haven't gone forwards and if we are looking to form an election-winning majority, we cannot be confident of that happening based on the results yesterday."
Labour leader Jeremy Corbyn described the results as "solid", saying the party had "consolidated" its gains in the 2017 general election.
He played down the significance of the failure to gain councils in London, claiming that Conservatives had deliberately talked up Labour's chances to "unrealistic" levels.
Labour came "within a whisker" of winning Tory flagship Wandsworth, he said.
The party gained seven seats in the south London borough once known as Margaret Thatcher's favourite borough, and Labour sources said they were just 141 votes short of taking control.
"In these elections we have won seats across England in places we have never held before," said Mr Corbyn.
"We won Plymouth from the Tories, who lost control of Trafford, their flagship northern council.
"And Labour has won even more council seats than at our high-water mark of 2014.
"The Labour Party is now well placed to fight and win the next general election - and form a government that will work for the many, not the few."
"We haven't in living memory run Wandsworth. We haven't in living memory run Westminster," he told World at One.
"Any electoral advance in those council areas... should be celebrated."
Mr Burgon said the last time the seats were contested in 2014 was a "high-water mark" for Labour, which gained more than 300 councillors under Ed Miliband.
"We've built on that success," he said. "Would I like us to do even better?
"Of course. But I don't think there's any need really for us to be considering it to be a failure.
"It leaves us in a position where we can win the next general election."