Hamilton was running fourth approaching the halfway point of the 61-lap race when he lost engine power, rapidly dropping back through the field before he was told to return to the garage and call it a night.
Starting from pole position, Vettel had no problems, leading all the way to take the chequered flag for the fourth time at the floodlit Marina Bay Street Circuit, despite twice losing the big advantage he had built up when the safety car was deployed.
Australia's Daniel Ricciardo finished second for Red Bull, his best result of a frustrating year, while Vettel's team mate Kimi Raikkonen crossed the line third.
Vettel's win was the German's third this season and the 42nd of his career, enabling the quadruple world champion to overtake the late Ayrton Senna into outright third place for the most lifetime wins, trailing only Michael Schumacher (91) and Alain Prost (51).
A proven master on the slow and twisty Singapore track, Vettel gave his rivals no chance when he planted his foot on the accelerator and made a flying getaway from the front of the grid.
The 28-year-old opened up a three-second lead over Ricciardo on the opening lap and was never seriously challenged even when the safety car was introduced.
Image credit: AFP
Its first appearance came on the 13th lap when Felipe Massa exited the pits and was hit from the side by Nico Hulkenberg who speared into a wall, wrecking his Force India car and ending his race.
The second came when a spectator walked on the circuit before climbing back though a hole in the fence.