Hamilton started from pole position, his first with his new team and the 27th overall of his Formula One career, but the 28-year-old Englishman had to settle for third behind the race winner Fernando Alonso and Kimi Raikkonen.
Even that position, however, was touch and go at the end of the 56-lap race as a charging Sebastian Vettel finished within 0.2 seconds of Hamilton.
Hamilton's ability to hang on to third gave Mercedes a driver on the podium in consecutive races for the first time since Nico Rosberg managed it in 2010. Hamilton had finished third in Malaysia when the team asked Rosberg to maintain his fourth-place position and allow Hamilton to move into third.
"It was a good race for me, and I'm quite happy with third," Hamilton said Sunday. "Of course, I would have liked to have won, but Fernando and Kimi were a little bit too fast for us.
"I was able to apply a little bit of pressure to Kimi, but not enough to get close to him and overtake, and then my tyres were shot at the end.
"I was trying to hold off Sebastian. I could see his car getting bigger and bigger in my mirrors, so it was nice to be able to hang on for the third place, so I'm happy with the podium, happy with the points."
Over the team radio, after the chequered flag, Brawn, the Mercedes team principal, was breathing a sigh of relief in seeing Hamilton hang on to third, conceding: "That was close."
Brawn added: "We are not quite there yet but we are not so far away, so let's keep working hard."
Mercedes appear to be slightly down in race pace compared to their quicker performances in qualifying.
Hamilton said: "The guys did a great job this weekend to get us where we were – on pole and finishing third. We got the absolute most out of the car we had and that's very satisfying, but like Ross said, we're not quite there yet in terms of overall pace.
"I'm not really sure where we're losing out, but everyone is working so hard and I know we can do it.There are a couple of areas we can focus on the car, but we also have to bring some more updates and keep on improving.
"I will also analyse a little bit and try to figure out whereabouts we're losing the time, see if we can zone in on that and try to improve there."
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