Tipsarevic will next meet Belgian wild card David Goffin, who rallied to beat Andreas Beck of Germany 4-6, 6-4, 6-2.
Second-seeded Nicolas Almagro also moved into the quarters after Steve Darcis of Belgium pulled out of the tournament due to a right shoulder injury. Almagro will face Japanese qualifier Yuichi Sugita, who beat Yen-hsun Lu of Taiwan 7-6 (5), 6-4.
"The fact is, playing in the quarter-finals in the first tournament of the year without playing a match is a handicap," said Almagro, after the top four seeds were given a first-round bye. "(But) I'm here to play my best. I think I'm ready, I've prepared well for the season."
A semi-finalist in the previous two Chennai Opens, Tipsarevic broke Bhambri three times in the first set and once in the second.
"It was a good start to the season, hope I can continue like that," Tipsarevic said.
Results on Thursday from the Chennai Open, a $398,250 ATP tour event on hard court at Nungambakkam Tennis Stadium (seedings in parentheses):
Singles
Second Round
David Goffin, Belgium, def. Andreas Beck, Germany, 4-6, 6-4, 6-2.
Nicolas Almagro (2), Spain, def. Steve Darcis, Germany, walkover.
Yuichi Sugita, Japan, def. Yen-Hsun Lu, Taiwan, 7-6 (5), 6-4.
Janko Tipsarevic (1), Serbia, def. Yuki Bhambri, India, 6-1, 6-3.