The rebellion today led their Liberal Democrat coalition partners to announce that they would not press ahead with a consultation on the closures.
The group would now face certain defeat at City Chambers if they tried to take the scheme forward.
The threat of closures for some of the Capital's schools is likely to resurface - but not until 2008 at the earliest.
City leader Jenny Dawe said: "This is very disappointing news. We entered into this process with a genuine motive to consult as widely as possible with families across Edinburgh.
"We have already gathered a number of good ideas which could have led to a very open and honest debate. This could have produced some very creative solutions for education in Edinburgh."
Education convener Marilyne MacLaren said: "This news is of considerable regret to me and my political peers. We embarked on this modernisation programme to ensure that we provide every child in Edinburgh with a first-class education.
"Although I appreciate that people had difficulty with some of the detail of the proposals, there was considerable support for the rationale behind the proposals from the educational community."
SNP leader and deputy leader of the council Steve Cardownie said today that too many schools had been named and called for officials to "go back to the drawing board".
It is likely the next hit list proposed will be significantly shorter. The current one included thirteen primaries, six nurseries and three high schools, as well as four community centres.
The SNP group, which was under pressure from its own MSPs to pull out of the consultation on the plans, met to discuss the issue last night.
Talks went on until midnight, with councillors finally agreeing to tell education leader Ms MacLaren today that she no longer has their support.
The news was met with elation from parents and pupils who had joined forces in a massive show of opposition to the plans.
Susan Russell, 30, of Restalrig Road, whose sons Mark, four, and Owen, three, attend Abbeyhill Primary School, said: "This is brilliant, great.
"People in Abbeyhill will be absolutely over the moon, the parents are going to be so happy."
Bernadette Brown, chairwoman of the Dalmeny Primary School parents committee, said: "It's brilliant news, absolutely fantastic.
"We have a meeting planned for today and we will still hold that to see what they've got to say to us. But people will be overwhelmed by this, it's the best news I could have had today.
"I'm sure this will be well received at every single school, everyone will be ecstatic."
Iain Kay, chairman of the Castlebrae Community High School parents committee, added: "At last the council has seen sense. The pupils will be very, very happy. There's not as much pressure on them now."
Consultations for each school had been approved by the council and the education committee, and the first public meeting on Princess Elizabeth and Cameron House nurseries was due to take place at Prestonfield Primary School on Thursday, September 20.
The SNP group has no way of stopping this going ahead but by withdrawing their support they have left the Lib Dems facing a certain defeat when the council comes to vote on the final proposals next year.
They will also face new lobbying from their coalition partners to ditch the plans completely.
Cllr Cardownie said: "We're not talking about the report coming back amended and then going back out again.
"We want to go back to the drawing board and look at the situation afresh. We have said this is not about savings and not about money, if that's the case the council can afford to take its time.
"Our main concern is that what we've heard from parents and pupils show they are lacking faith in the process and therefore they will have no faith in the outcome."
The group also feared some of its councillors would bow to public pressure in their wards and vote against the administration to save local schools.
"Some parents have been in contact with parents, attending meetings, and reporting back to us. Clearly the situation is untenable," he added.
The SNP group's education spokesman David Beckett said there were also educational reasons why the plans should be scrapped.
He said: "There are no references to the school closure guidelines set out by Peter Peacock [former Scottish Education Secretary] in 2004. We are not confident the research to support the council's optimum school sizes [of 400 in primary and between 900 and 1200 in secondary] is robust.
"And we feel it does not strengthen the SNP commitment to have maximum class sizes of 18 in P1 to P3."