Running sixth by lap five, Hamilton then passed Kevin Magnussen's Haas a lap later, then Gasly, and moved into fourth by lap eight.
Raikkonen attempted to leave a pit stop before all four of his tires were safely attached and his rear left wheel hit a member of the crew, with video showing the mechanic's leg buckling horribly.
In reality, Vettel's soft tyres were becoming a great hinderance, as the turns on the final laps had to be progressively taken with less and less speed to make sure the vehicle would stay on the line.
Hamilton expressed his annoyance over team radio as he came into the pit lane after P3, saying spectators got too close to his Mercedes.
Verstappen had a more frustrating race.
In one thrilling maneuver during the race Hamilton overtook three drivers in one go. The German driver won in Melbourne his 48th career win and 100th podium with Hamilton finishing second after his team made an error of judgment during Vettel's pitstop.
Both Bottas and Hamilton not only managed to match the Ferraris for pace on their Mediums, but also were going marginally quicker.
Vettel now leads the championship standings with 50 points; Hamilton, with 33 points is second, and Bottas, with 22 points is third. As the team tended to the fallen crew member, Raikkonen brought his auto to a stop before clambering out, ending his race on a sour note.
Valtteri Bottas tried to catch Vettel but fell short by just 0.699 seconds.
US, North Korea in Secret, Direct Talks
Once a location is agreed upon the officials said that the date will be set and the agenda discussed in greater detail. South Korea brought Kim's offer to meet to the White House last month, which Trump reportedly expected on the spot.
Mercedes' decision to go for the more durable medium tyres and a one-stop strategy, compared to Ferrari's move to bolt on softer rubber in anticipation of a two-stop race, brought the evening alive.
The injured mechanic has been taken to hospital for treatment.
Meanwhile, Vettel was still leading from Bottas as the first round of pit stops began.
"I feel really pleased". A mistake at Turn 1 allowed Bottas to further close, setting the stage for a grandstand finish.
Another area in which Liberty has big ambitions of expanding is OTT, yet the company's much-vaunted plan to launch Formula One's new live streaming service, F1 TV, have stalled once again after a push to roll out the product in certain countries before the season-opener in Australia were put on hold.
Honda enjoyed its best result since returning to the championship after rookie Pierre Gasly took fourth in just his seventh grand prix, and Marcus Ericsson claimed his first points finish since 2015.
Carlos Sainz Jr. finished 11th for Renault after suffering on his Soft tyres, having completed 40 laps on the set he crossed the line on.
BC-CAR-F1-Bahrain GP, 3rd Ld-Writethru, 811 Vettel wins Bahrain, Raikkonen's auto breaks mechanic's legAP Photo XLB112, XLB115, XLB109, HAS135, HAS136, XLB108, XLB107, HAS128, XLB118Eds: Adds details, quotes, photo links.