A late charge from Hamilton wasn't enough to bring him into the battle for victory as he finished six seconds back from Vettel in third place. Hamilton finally stopped a lap later, swapping his soft tyres for mediums.
Cigorini fractured his tibia and fibula in his left leg and was taken to hospital immediately, while Raikkonen was disqualified a lap, as Ferrari were fined €50,000.
"This grand prix confirms that the Scuderia has a great vehicle, a solid team and two drivers on top form".
Esteban Ocon opened Force India's account for 2018 with a hard-fought world championship point in 10th.
Vettel now leads the championship standings with 50 points; Hamilton, with 33 points is second, and Bottas, with 22 points is third.
"They weren't getting any better".
Contact between Sergio Perez and Brendon Hartley at Turn 4 saw both drivers' races ruined as Perez was forced to pit on the first lap and Hartley was hit with a ten-second penalty for causing the collision.
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Video footage of the fracas, which caused a "minor injury" to one of the passengers, has since gone viral on social media. He did, however, fight and lose to Floyd Mayweather Jr in a money-spinning boxing match in August 2017.
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Team-mate Luke Rowe later withdrew after being caught up in a crash involving Tony Martin and Alexander Kristoff with 48km to go. Belgium's Michael Goolaerts was taken to hospital after crashing earlier in the race.
In one thrilling maneuver during the race Hamilton overtook three drivers in one go.
Magnussen was quickly passed for P5, which turned to P4 once Gasly was dealt with.
Having started on soft tyres, Hamilton was always likely to consider a one-stop strategy, although the expectation was that Ferrari would opt for a two stop.
The unsafe release from the pits was key flashpoint in the race won by Vettel, who made it two wins out of two in his 200th race. [But it was] mostly the fact I had a better feel with the vehicle.
Raikkonen, who was pitting for a second time, was not at fault for the collision. "He is now recovering with the best assistance", Ferrari tweeted. First we'd have to get past Lewis and then chase Valtteri and pass Valtteri.
The one-second gap was breached by Bottas with two laps to go, allowing him to utilise DRS to close up on the back of Vettel.