He added that he was proposing to President Alexander Van der Bellen that Foreign Minister Alexander Schallenberg, a career diplomat backed by Kurz's party, that he take over as chancellor.
Greens leader and Vice Chancellor Werner Kogler indicated he was satisfied, meaning Mr Kurz had succeeded in pulling their coalition back from the brink.
"I believe this is the right step for future government work," Greens leader and Vice Chancellor Werner Kogler said in a statement, adding that he had had a "very constructive" working relationship with Mr Schallenberg.
A star among Europe's conservatives and known for his hard line on immigration, Mr Kurz, 35, became one of the continent's youngest leaders when he took over as chancellor in 2017 at the head of a coalition with the far-right Freedom Party.
Prosecutors said on Wednesday they had placed Mr Kurz and nine others under investigation on suspicion of breach of trust, corruption and bribery with various levels of involvement.
Starting in 2016 when Mr Kurz was seeking to take over as party leader, prosecutors suspect the conservative-led Finance Ministry paid for advertisements in a newspaper in exchange for manipulated polling and coverage favourable to Mr Kurz.
Mr Kurz has pledged to defend himself against what he says are false allegations. He had said he was willing to keep governing with the Greens. But the left-wing party said the investigation made Mr Kurz unfit to serve as chancellor and called on his party to name a successor who was "beyond reproach".
Austrian chancellor Sebastian Kurz resigns over allegations of corruption
The Greens began talks on Friday with Austria's three opposition parties, which have all demanded that Mr Kurz resign and had planned to submit one or more no-confidence motions against him at a special session of parliament on Tuesday.
Austrian media reports before Mr Kurz's announcement had said he would step down only temporarily. While Mr Kurz did not say that he did say he would mount a legal defence: "Above all ... I will of course use the opportunity to refute and disprove the accusations that have been made against me."
"Is it enough?" the leader of the liberal Neos party, Beate Meinl-Reisinger, told a news conference soon after Mr Kurz's statement, referring to the seriousness of the accusations against Mr Kurz.