A blunder by the Red Bull pit crew at Mark Webber's first stop, as a nut fell out of the socket, led to a delay and put him on the back foot in his chase for a top-three finish.
It prompted Red Bull to switch to an unusual three-stop strategy at the Yas Marina Circuit that only called for two, but despite a host of fast laps late on, the Australian was unable to make it work.
Stopping for his final set of tyres at the end of the penultimate lap elevated Button onto the podium for the seventh time in the last eight races.
Given the 31-year-old had been plagued by an intermittent issue with his KERS power-boost system, the end result was welcome.
"I was very surprised by their tactics," said Button.
"I don't know if they thought I had the KERS problem for the rest of the race because when I did I was obviously a lot slower and he was pushing me around for 15 laps.
"But as soon as I sorted the problem out, I would go half a second quicker. I don't know.
"It was a strange decision to make, but they also made a mistake on the first pit stop which lost them a lot of ground.
"That really hurt him and he wasn't able to really challenge after that.
"Given what happened with the KERS it was a pretty tricky race, so to get to the end and be on the podium, I'm pretty happy with that."
Finishing third means Button only yielded 10 points to race winner and McLaren team-mate Lewis Hamilton, guaranteeing his place as the best of British within the team this year.
It is the first time since Hamilton began single-seater racing in 2001 that he has been beaten by a team-mate.
Button was at least complimentary towards his compatriot following a faultless drive to the line after double world champion Sebastian Vettel retired due to a puncture sustained early on lap one.
"I just want to say 'fair play to Lewis'. He did a great job," said Button.
"It's very positive to move towards the end of the season on a high because it's the best possible way to build for the new season."