The world number two managed to break back immediately on both occasions before holding for the first time, and then used her powerful groundstrokes to dominate the longer rallies and claim the opener.
With her serve firing, Kvitova charged out of the blocks in the second set, winning the first three games to spark what looked to be a stroll into the quarter-finals and breaking Azarenka's serve again to open up a 5-1 lead.
But the Czech double faulted on match point, allowing Azarenka to break back and mount what was almost a fightback for the ages.
"I don't know why (the double fault) happened. I really wanted to finish it earlier but unfortunately I had to fight in the second set and I did quite well in the tiebreak," Kvitova said courtside after her win.
Azarenka, her confidence restored, pinned Kvitova back behind the baseline with her aggressive tennis, and broke again to level the set at 5-5 and force a tiebreak.
Kvitova hit 40 winners to Azarenka's 14, but she also made 33 unforced errors compared with 17 from her opponent.
The Czech's composure came to the fore in the tiebreak and she closed out the match with a powerful backhand down the line to set up a quarter-final clash with either Russia's Veronika Kudermetova or Croatia's Donna Vekic.
(Reporting by Simon Jennings in Bengaluru, editing by Pritha Sarkar)
