It was the first time the seven-time world champion had scored a rostrum result since returning to the sport with Mercedes in 2010.
Afterwards, Schumacher admitted that he had not realised he had taken third sport until after he crossed the finish line.
"I didn't actually think about a podium at all, neither in the end of the race, it was in crossing the line when I asked my guys where we finished," he explained.
"I saw [Mark] Webber's pitboard and close to the end it showed him eighth and seventh and knew I was one place ahead of that one. And then the boys told me third, that's a podium - and I thought I can't believe that.
"It is something I didn't expect, you sort of lose count because I was pretty busy in those last stages of the race plus, [Lewis] Hamilton and [Pastor] Maldonado had their issue, so I lost counting at that stage."
Schumacher added that experiencing such emotions was the driving force behind his decision to return to F1 and that achieving third in such a dramatic race only made the result sweeter.
"It is those moments that definitely you enjoy deeply and it's just a wonderful feeling to be back after such a long time," said Schumacher, after a 99-race wait to return to rostrum – the last time he stood on one was following his victory in the 2006 Chinese GP.
"We were a couple of times close to it, and finally it happened in a very spectacular way and on a track where it is difficult to pass – but due to various strategies and differences on track it was pretty exciting and that is best way if you manage the way I did today to jump on the podium.
"To do it in an exciting way makes me extra happy.
"But that is what I am here for, to be excited, hearing finally the message P3 is a sweet finish to an unexpected race. Converting from a one-stop to a two-stop strategy meant we had good tyres at the end."