Set a total of 202 to claim victory, the hosts clinched the win with two balls remaining as MS Dhoni stroked Shane Watson's full toss through cover for a pair of runs.
Dhoni (24 n.o.) was part of a winning century partnership with man-of-the-match Yuvraj Singh, who managed 77 off 35 balls.
India had been facing a run-rate of almost 12 when Yuvraj entered the fray at 100/4, but the cancer survivor and Dhoni blasted two runs a ball as they bludgeoned their way to victory.
Yuvraj, who underwent treatment for lung cancer in 2012 and is making a second comeback after suffering a back injury this season, said a series of India A matches helped him regain full fitness.
"I shed a few kilos and worked really hard in the off season," he said. "I came with a fresher mind. I need some really good games to come back to action, (India) A games were a pretty good platform for me.
"I'm very confident when Mahi is around, he is one guy who goes after the bowling. I just thought I would try and take the game to the end. It is always a pleasure and honour to be playing in the Indian jersey."