The 2008 Formula 1 world champion had struggled to match team-mate Nico Rosberg in India and Abu Dhabi.
Mercedes discovered a crack in Hamilton's chassis after the latter race and provided a new one for Austin.
Hamilton qualified fifth and finished fourth in America, to his delight.
"The car was night and day different compared to where it was in Abu Dhabi, so I'm sure the new tub helped," he said.
"I'm really just so happy that I could contribute and get some points for the guys. They've been working so hard all year and they've had so many difficult races in this half of the season.
"To finally have a half decent qualifying, move forward, utilise the tyres and have life left in them, makes me really, really happy."
Fernando Alonso's Ferrari closed in on Hamilton late in the race, but the Briton was able to respond.
"I could see Fernando was catching, so I was just looking after the tyres and waiting until he started to push," he said.
"He started to push, so I reacted a little bit.
"When he was going to bring out a tenth, I was going to bring two tenths. When he was going to bring three, I was going to bring four.
"I'm really, really happy I could do that today."
Hamilton result moved him past the sidelined Kimi Raikkonen to take third in the drivers' championship, though he could still be beaten to the place by Mark Webber in Brazil next week.
The Mercedes driver said he was not too fussed about a top-three points position anyway.
"That doesn't mean anything, plus the guy that was in third wasn't even racing," said Hamilton.
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