The Ferrari driver took his second win of the season after jumping Lewis Hamilton for second at the first corner, and getting ahead of Valtteri Bottas for the lead by being the first frontrunning auto to make a pitstop.
Vettel is chasing his fifth world title and is seven points ahead of three-time champion Hamilton in the drivers' standings, while Ferrari is three points in front of Mercedes in the constructors' race.
After claiming the first pole position of his career, Bottas said the fight for the championship was now a three-way battle, but the Finn will be desperately disappointed here after he finished a distant third.
Hamilton also finished second to Vettel in the Australian opener, before winning in China, and he said that defeat had not been due to any "massive fault of my own" but rather due to heavy tyre wear.
Vettel now has 68 points to Hamilton's 61.
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"I don't know why it would make me hungrier", Raikkonen said. "But for now the team has done a really, really great job". The vehicle was really unbelievable to drive and I could control the race in the end.
"We tried to maintain the order and not interfere in the running order because it's an very bad call you need to make", Wolff added. Kimi Raikkonen took fourth for Ferrari after a poor start ahead of Daniel Ricciardo, who was in the mix at the start but lost ground when he switched to soft tyres for the second stint. "The factory has come alive and hopefully the success in the last few races helps us". Losing points for a team, particularly when you could have won the race, is definitely painful but I gave it everything I could.
"After a couple of laps, I was on Valtteri's gearbox for the first stint and not falling back too much". The incidents were coming fast, with 18-year-old Lance Stroll hit from the side on lap 13 by Carlos Sainz Jr. - forcing the safety auto to come out.
Vettel came in for new tires on lap 11, prompting Red Bull's Max Verstappen to pit on the next lap.
Both Mercedes drivers used the safety vehicle incident to pit for new tires, but Hamilton made the mistake of holding up Ricciardo.
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The five-second penalty would hang over Hamilton for the rest of the race and hamstrung his chances of mounting a fight back.
When the race began in earnest again on lap 17, Hamilton burst straight past Ricciardo into third while Vettel brilliantly defended the corner twice to fend off Bottas. Vettel was braver on the brakes and passed Hamilton around the outside.
Valtteri Bottas says he understands why Mercedes ordered him to let teammate Lewis Hamilton pass at the Bahrain Grand Prix, but admits he did not like doing it.
On a significantly cooler day, with a strong wind, the temperatures fell to 25 for the air and 30 degrees Celsius shortly before the race when Belgian Stoffel Vandoorne withdrew after yet another engine failure with his McLaren Honda, his third of the weekend and the team's fourth.
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