Kimi Raikkonen and Max Verstappen both crashed out at turn one while Vettel sustained significant damage to his Ferrari in a dramatic start to the race in very hard conditions at the Marina Bay Circuit.
It all played perfectly into the hands of Mercedes driver Hamilton, who went from fifth on the grid to first and held off Red Bull's Daniel Ricciardo to clinch his third win in a row. "I don't want to catch any of that stuff", he said. "Fantastic job with the team strategy", Hamilton said over the radio, before leaping out of his auto and hugging his engineers.
"I needed it to rain, and as soon as it rained, I knew where I was going to finish", Hamilton said.
"I've got no idea -- and that is the concern I've always had with this race", Romain Grosjean told Autosport.com earlier, when asked if it would be safe to race in the rain. "It was really unfortunate for the Ferraris but a great result for the (Mercedes) team".
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Hamilton and Mercedes had braced themselves for a race of "damage limitation" after struggling in qualifying.
This is the first abandonment of the season for the German. "The quickest auto and the quickest driver were out within a minute into the race and that can happen all the time".
The win takes the Brit 28 points clear of Vettel, who was blamed by Max Verstappen for causing the crash, and a further 51 points ahead of team mate Bottas ahead of the next race in Malaysia.
Raikkonen and Red Bull's Max Verstappen collided on the run down to Turn One before the Finn plowed into the side of Vettel.
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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.
While the three leaders were out on the spot, Alonso tried to continue after the resulting safety auto, but his McLaren was too damaged and he had to retire.
Having built on that advantage, championship leader Hamilton was frustrated by the reappearance of the safety auto on lap 11 when Daniil Kvyat drove into the barriers.
Nico Hulkenberg and Kevin Magnussen both retired late in the race to bring the total number of casualties to eight.
Race - Hamilton wins dramatic wet-dry Grand Prix in Singapore
I'm happy that not only I retired, but all three of us, so we all have a bit of pain. "That's a seriously healthy gap". The collision left Vettel's auto with a hole punched in the side and he spun into the wall at turn three.
The FIA has said the incident will be investigated after the race.