Raikkonen was given the green light to go but it was an unsafe release and, as he pulled away, he hit one of the mechanics.
Toro Rosso-Honda's Pierre Gasly took a brilliant fourth place.
Vettel held off the cost of Mercedes' Valtteri Bottas to take victory in his 200th grand prix, however the event was soured after a member of Raikkonen's pit crew was taken to a medical heart after an incident throughout lap 36 of the race.
Vettel made a clean start and Bottas swept past Raikkonen to split the Ferraris on the opening lap.
Moments later, there were heads on hands in the Red Bull garage as Ricciardo's race was ended. A Virtual Safety Car was deployed. The race was fought on a lethal mixture of pace and strategy, that too between many teams and it probably helped that there were three quick drivers out of their regular starting positions - Lewis Hamilton, Max Verstappen and Romain Grosjean (yes, the Haas is quick this season!).
Red Bull team boss Christian Horner agreed it had been a "brutally frustrating" race.More news: Transgender weightlifter Hubbard injured in competition
World champion Lewis Hamilton could only manage fourth, and will start from ninth on Sunday after incurring a grid penalty. Despite being on soft tyres while those around him were sporting supersofts, he was soon on the march.
Kevin Magnussen bounced back from his disappointment in Australia to finish fifth for Haas ahead of Renault's Nico Hulkenberg.
Hamilton meanwhile came from ninth, after a five-place grid penalty, to secure a record-equalling 27th successive points finish. That triggered Gasly, Hulkenberg and Ocon to come in next time by. Hamilton was brought into the pits for his one and only pit stop at the end of the lap, also switching to mediums that would take him to the chequered flag.
Ocon rounded out the top ten just ahead of his Force India team Sergio Perez, who had lost time after being tipped into a spin by Brendon Hartley on the opening lap.
"With Bottas' pace I thought he would catch me".More news: Commonwealth Games: Jitu Rai claims gold, Mehuli Ghosh bags silver
All that makes it 2:0 for Ferrari over Mercedes.
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.
Magnussen dropped to 10th when he came in for his second stop and wasn't impressed with his team-mate Romain Grosjean.
This is effectively McLaren's second home race given their financial ties with the Bahraini royal family, but the guests which squeezed into the team's hospitality suite were provided with little fanfare as Alonso qualified a torrid 13th.
Ferrari totally botched Raikkonen's second tire change.
Cigarini was knocked over when Raikkonen was released before his rear-left tire had been fitted at the Sakhir circuit.More news: Centre advises states to increase security for Bharat Bandh on Tuesday
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