The crash gave victor Lewis Hamilton a 28-point lead in the title race.
Daniil Kvyat put his Toro Rosso into the Turn 7 wall amid tricky conditions on Lap 11, and Marcus Ericsson crashed his Sauber on the Anderson Bridge on Lap 38.
"I think there was damage on the auto already", he said of his second incident.
Vettel had got around the first corner ahead, but then spun, and with his vehicle badly damaged he retired from a race for the first time in 2017.
Remarkably, that gave main title rival Lewis Hamilton - whose prospects of a strong race looked bleak in the dry when he qualified fifth - the lead of the race as the Safety Car came out while the damaged cars were removed from the circuit.
"For sure, it's bitter but it's done", said the four times Singapore GP victor, whose retirement ended a run of 18 successive points finishes.
"I was really waiting for the opportunities".
The incident allowed Hamilton into a lead he would keep through two more safety vehicle periods and a perfectly-timed switch to dry weather tyres.
Formula One's Lewis Hamilton said a TV documentary had inspired him to go vegan to improve his health and avoid harming the planet.
"Guys what can I say - what a turnaround today!"
"By letting people that are following me know, maybe I can encourage a couple of people to do the same thing". "I'm a little bit disappointed to miss out on the win".
"It was going to be one place maybe or hope for reliability for another auto but for the rain to fall, I was so happy, you can't imagine how happy I was".
"It's bitter but it's done", said the German, a four times Singapore GP victor, whose retirement ended a run of 18 successive points finishes and left him with a mountain to climb. However, race control decided against taking the option of starting the race under the Safety Car.