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.
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"It must be one of my best races. It was pretty intense," Vettel said. "Lots of pressure from behind; Daniel had a very good race looking after his tyres.
"Second stint I was dictating the pace. From then, I was trying to control the gap. We had a really great weekend."
Nico Rosberg finished fourth for Mercedes to cut Hamilton's championship lead to 41 points with six races remaining.
Vettel trails Hamilton by 49 and Raikkonen is 145 adrift with a maximum of 150 available. No other driver has a chance of winning the championship.

Mercedes AMG Petronas F1 Team's German driver Nico Rosberg during the Formula One Singapore Grand Prix in Singapore

Image credit: AFP

"If we have more weekends like this, yes we can catch Mercedes," Vettel said.
"Massive attack. Maybe we can make the impossible possible - we will definitely go for it."
[LEWIS LOSES: Lewis Hamilton forced to retire in dramatic Singapore Grand Prix]
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.

Massa

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.
[CRAZY: Spectator wanders along track during Singapore Grand Prix]
"If there was no safety car we could have been close for the undercut but the two safety cars came at pretty crucial times," Ricciardo said.
"It was really good to back up qualifying. We matched Seb in race pace and we have got to be pretty happy."
Hamilton, who needs one more win to draw level with his childhood hero Senna, endured his worst weekend of the season.

German Ferrari driver Sebastien Vettel during the Formula One Singapore Grand Prix in Singapore

Image credit: AFP

The Englishman struggled with his tyres in qualifying, starting fifth on the grid, and his race came to an end just when it looked like he might be able to challenge for a spot on the podium.
"I was feeling super optimistic and easily keeping up," he told the BBC. "I was hoping for a quick fix but it never came."
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