After starting from ninth - the result of a disappointing qualifying and a five-place grid penalty – Hamilton lost ground at the start and then struggled for grip and performance.
His fortunes reversed following his second stop however and he surged back up the field, eventually sealing fifth with a last-lap pass on the Red Bull of Mark Webber.
"I'm very, very happy - massively happy," Hamilton said.
"The race started so badly. I wanted to get a good start, but I got a terrible start. Then I was so slow. I was nowhere, no speed at all.
"No matter what I did to the car, it didn't get any faster.
"All of a sudden something happened after the second pitstop; all of a sudden the car started reacting differently and I could push again. It must have been something that happened to the car in qualifying when we changed the gearbox, and it sorted itself.
"At the beginning I was a sitting duck, and then it was great to be able to do some overtaking. All of a sudden, the car was amazing."
ROSBERG: RACE WAS 'SO BAD'
Team-mate Nico Rosberg, who had started from a surprise pole, said Mercedes needs to work on translating its Saturday performances across into the races.
The German fell back almost relentlessly during the race and, despite some dogged defending and fights, eventually came home ninth.
"It's been an unbelievable day - so bad," Rosberg said.
"I started strong and I was happy in the podium position but I knew it would be difficult [to stay there].
"Going backwards all day long was not a good feeling. Obviously we were quickest yesterday so something has to be right, but we need to shift it more to the race."