The SNP had been hoping to topple Labour from the Glasgow stronghold where it has held power for more than three decades.
The election results showed the SNP in the ascendency, with the Tories and Labour "scrapping for second place", she said.
This leaves an interesting challenge ahead, whether the "old guard" coalition of a Labour/SNP administration will prevail, or whether the Conservatives can come to a working agreement with the two Indy councillors.
"More seats, more votes, more councils won, bringing almost 40 years of Labour administration to an end and becoming the largest party in all of our major cities. They will fight against Tory attempts to drive down living standards and will not be distracted by campaigning for a divisive second independence referendum". The SNP won 32.3% of the vote, while the Conservatives secured 25.2% and Labour 20.2%.
Despite topping the poll, the party lost overall control of Dundee and Angus councils, which it had before the election.
But even if everyone who voted independent on Thursday would have voted SNP in a general election (a highly improbable supposition in itself), that clearly can not account for all of the difference between the two performances.
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.
"Every Labour councillor will always put their local communities first".
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Speaking in Ellon as he launched his campaign to be re-elected on June 8, Mr Salmond stated: "I said the Tory support was on the rise, it should be said in Gordon constituency we got more councillors in than the Tories, we got 11 councillors to their nine and six for the Liberal Democrats".
Meanwhile, the SNP have not had any difficulty increasing their vote in local by-elections during the last two years; on average the party's share of the vote has been up ten points on 2012 in such contests. "If there's a message at all on that issue, it's a message for the Conservatives".
The projections were calculated based on first preference votes cast at last week's local elections, which the party believes is likely to reflect voting patterns on 8 June.
He also reckoned Ian Murray, Labour's sole remaining Scottish MP after the 2015 General Election, could also hang on in Edinburgh South.
On a campaign stop in Skegness on Saturday, Nuttall claimed the results were not a disaster. "Ukip does have a great future".
"The Tories are the party of the "rape clause" and hard Brexit".
"She will buckle", Nuttall said. Fisheries will go, there will be movement on immigration, on freedom of movement - she will buckle on that.
Other deals may be harder to stitch together, not least because of the SNP's declared refusal to work with the Tories. "Ukip could be bigger than it ever was before".