A spokesman for the Scottish Conservatives said the results proved it was the only party capable of challenging the SNP across Scotland in the general election.
Sturgeon has explicitly ruled out doing any deals with the Tories, even though the SNP had two coalitions with Conservative councillors after the 2012 election.
The Scottish Liberal Democrats received a 6.8% share of the first preference votes as they won 67 seats, with the Scottish Greens winning 19 seats on 4.1% of the votes.
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".
"So they put that issue centre stage and they lost the election".
She added: "Let's take the Tory argument at face value".
Similarly, Labour would be attacked by the SNP if local politicians struck up partnerships with the Conservatives.
"We won the local government election is Gordon this week, beating the SNP into second place".
SNP leader and First Minister Nicola Sturgeon pointed out that her party had ousted Labour from control of a Glasgow City Council after 40 years, and still had the largest number of councillors across the country.
She said: "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". Not only is the Tory revival real (some Nationalists appeared to be in denial about that), but if some of the swings on Thursday are replicated in Westminster constituencies next month then the SNP could lose some senior figures.
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Jon Howard, a Phoenix police spokesman. "Our hearts go out to the surviving families", Phoenix police chief Jeri Williams said. Lydia Lopez, the mother of Jesse Olivas , said she suspected her son's death was connected to the Serial Street Shooter .
She added: "The truth is the more Tory MPs Westminster has, the heavier the price Scotland will pay".
"We need a strong Scottish Parliament, delivering on the powers we already have, and a voice at Westminster that won't put up with Tory attempts to hold Scotland back".
One user wrote: "Time for her to leave and let Scotland prosper in a strong union".
And although the SNP's share of the first preference vote may have held steady across Scotland as a whole as compared with 2012, it has not held up everywhere.
"The Tories are the party of the "rape clause" and hard Brexit".
"In terms of the strengthening of Theresa May, certainly if you look at the results in England, Theresa May is on course for a landslide victory - that raises a really important question in Scotland over the next few weeks".
"Scotland must speak with a clear voice in this election to stop the Tories in their tracks".
British Prime Minister Theresa May's Conservative Party has opened up a 22-point lead over the opposition Labour Party ahead of a June 8 national election, an ICM opinion poll for the Guardian newspaper showed on Monday.
Labour also had a higher percentage of first preference votes than the SNP in Coatbridge, Chryston and Bellshill, up 1.65% percentage points, and was 0.91 percentage points higher in Lanark and Hamilton East.