Results showed a loss of 144 seats for Labour - only five weeks before the snap general election.
And he said that if Labour won, there would be "a reckoning" with big businesses and bankers who had stripped industrial assets, crashed the economy and ripped off consumers and workers.
Shadow chancellor John McDonnell said that Labour's vote "held up" in areas of Wales where Jeremy Corbyn had campaigned, and said there were signs that voters were developing more confidence in the party leader as he had the chance to get his message across.
The main opposition Labour Party had won 609 seats, a net loss of 202, and controlled five councils, all in Wales.
The victories, fuelled by a collapse in the Ukip vote as anti-EU supporters flocked back to the Tories, mean Theresa May could see a substantial increase in House of Commons majority if repeated in June.
Too many fantastic councillors, who work tirelessly for their communities, lost their seats.More news: China launches its 1st home-built aircraft carrier
May has asked voters to back her leadership of the country and her plans for negotiations with European Union leaders.
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".
The party has so far won just one seat.
Theresa May is viewed much more favourably than the Conservative Party.
The local government elections in Britain this week have taken on a particular significance in terms of providing an indication of voting intentions ahead of the Westminster poll in just over a month's time.
British Prime Minister Theresa May maintained her strong lead in opinion polls ahead of next month's national election, with one analyst saying she was on course for the kind of huge success Margaret Thatcher enjoyed over 30 years ago.More news: Some 2000 People Evacuated in Canada Due to Heavy Flooding
It became clear early on that the Conservatives were set to be the big winners in the local elections.
Adam Drummond, head of political polling at Opinium, said the fall-out from the Brexit result was proving less damaging to the Tories than to other parties.
Pollsters got the result of the 2015 election badly wrong, concluding that they had underestimated support for the Conservatives and had over-represented Labour supporters in their surveys.
At the time the Star went to print, Lib Dems had 402 councillors, a net loss of 28; the Scottish National Party had 365, adding 31; Plaid Cymru was on 202, a net gain of 26 and the Greens were on 40, a net gain of six.
But remember that this number isn't the same as a prediction of what will happen in June.
Stephen Kinnock, who is contesting the seat of Aberavon, said the results amounted to a "disastrous picture" and Corbyn's leadership had come up as an issue on the doorstep. But its overall vote share of 18 per cent was their best result in any election since entering a coalition with the Tories in 2010, and put them nine points behind Labour on 27 per cent.More news: Nigerian leader meets with 82 Chibok schoolgirls
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