Hamilton controlled the race from the front but was under instruction from Mercedes management not to try any strategic games to delay Rosberg.
However, he disregarded message after message to speed up and prevent Vettel and Red Bull's Max Verstappen from closing on Rosberg.
The German in turn complained on the radio to the team to "consider doing something" and the tense title battle will do nothing to improve the cool relationship between the two drivers.
At one point, as Vettel homed in on Rosberg in the closing laps, Mercedes executive director (technical) Paddy Lowe, who only speaks to the drivers on the radio in extreme circumstances, told Hamilton: "Lewis, this is Paddy. We need you to pick up the pace to win this race."
Hamilton replied: "I'm in the lead right now. I'm quite comfortable where I am."
But his tactics were not enough and Rosberg whooped with delight as he crossed the line and, after wife Vivian congratulated him over the radio, did doughnut spins on the pit straight before climbing on to the nose of the car to take the cheers from the crowd.