The Briton faces the prospect of trying to overhaul a 36-point deficit in the remaining 18 races of the season.
Ferrari`s Sebastian Vettel came second despite colliding with team-mate Kimi Raikkonen on the first lap.
The incident damaged both cars, with Raikkonen fighting back to fifth behind the Red Bulls of Daniil Kvyat and Daniel Ricciardo.
Raikkonen`s rise through the field included passing Hamilton in the closing stages, Ferrari`s decision to use soft tyres for the final stint as opposed to the mediums chosen by Mercedes giving the Finn an advantage as the Briton fought in vain to try to pass the Williams of Felipe Massa.
Hamilton will express his confidence at being able to erode Rosberg`s points lead and will expect his awful luck of the first three races to turnaround - but the fact remains that no driver who has won the first three races of a season has failed to go on to claim the World Championship.
That crazy first lap
The race was shaped by an incident-packed opening, with a number of collisions on the first lap leading to the introduction of the safety car on lap four.
By that stage, Hamilton and Raikkonen had already pitted once to replace front wings damaged in first-corner collisions.
Raikkonen moved right into Vettel`s path as the German moved left to ensure he was not tagged by an aggressive dive down the inside by Kvyat.
Hamilton was then turned into by Sauber`s Felipe Nasr as he avoided Raikkonen`s damaged car swerving back on to the circuit.
Vettel confronted Kvyat about the incident in the pre-podium room after the race, saying: "If I don`t go left, you crash into us and we all three go out - you are like a torpedo."
Kvyat replied: "We didn`t crash." Vettel responded: "You didn`t. You were lucky this time." Kvyat shrugged: "I`m on the podium; you`re on the podium."
In the post-race interviews, Kvyat added: "You see the gap, you go for it on the inside. You see one car it is hard to see both. It was a risky move, I agree with Seb but you have to take risks and I am on the podium."
Ricciardo`s fizzing start turns flat
At the front, Ricciardo had made a stunning start to take the lead from pole-sitter Rosberg at the first corner.
In the process his left-rear wheel had been tagged by Rosberg`s front wing. Whether it was that or the debris on the track, Ricciardo suffered a left-rear puncture on lap three, allowing the Mercedes to pass.
Ricciardo would almost certainly not have been able to hold off Rosberg in any case, but his removal from the lead left the German serene out front.
Mercedes decided not to stop him under the safety car and when the race re-started on lap nine, Rosberg simply drove off into the distance.
Just the five pit stops then, Lewis...
Hamilton, meanwhile, was in the wars. Mercedes stopped him twice under the safety car, doing only one lap on the super-soft tyres in an attempt to ensure he could use only the softs for the rest of the race.
So Hamilton had done three stops within the first six laps and he was lying third from last at the restart.
He began his fightback from there, with the Ferraris doing the same - Vettel starting from 11th and Raikkonen not far behind Hamilton, who said his progress had been affected by damage to the car in the incident with Nasr.
"The car was pretty damaged," said the 31-year-old afterwards. "I`m sure it was some aero components but I think it was the suspension as well because it was flexing all over the place."
Mercedes F1 boss Toto Wolff said the front wing had damaged the leading edge of the floor and the barge boards when it came off and that the car had lost a "lot of aerodynamic performance".
After a further switch to the medium compound tyres to last the race, Hamilton was not able to recover as much ground as the Ferraris and faces an uphill battle if he is to stop Rosberg winning his first world title this year.
