Mark Serwotka, General Secretary of the Public and Commercial Services Union PCS.L, told reporters progress had been made in talks with ministers over looming job cuts.
The planned strike on Thursday, the day before the opening of the Games, had been roundly condemned by the government.
"People have their grievances but surely the day before the opening ceremony of the Olympics is not the day to do it," sports minister Jeremy Hunt told BBC radio.
Interior ministry lawyers were due in court at midday to seek an order stopping the action on the grounds the strike ballot of PCS members might not have been conducted properly.
"We believe that significant progress means that there is now no case for the union to proceed with industrial action tomorrow in terms of the one-day strike that was planned," Serwotka said.
Serwotka said it was a promise of new investment and 1,100 new jobs, confirmed in talks barely 24 hours before the strike, that had persuaded them to call off the action.
He admitted there had been a public backlash against the strike plan on the eve of the Games, something he blamed on "scandalous" comments from ministers and Conservative lawmakers.
But he denied that was why they had called it off.
"The abuse and vitriol that has been directed at us has been without recent precedent," he said.
"To have hard-working civil servants have their patriotism questioned personally I think is sickening. (They) do not need their patriotism questioned by failed politicians like Jeremy Hunt." (Reporting by Michael Holden; Editing by Sonya Hepinstall)
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