Bhambri was dominating 16th-ranked Fognini when he conceded the match with a left quadriceps strain when Bhambri was leading 6-1, 5-5. Fognini quit just when he seemed to be finding his rhythm.
Bhambri, ranked 195th, became the first Indian to reach the Chennai quarters since 2009.
"It's great to be in the quarterfinals of an ATP tour event before the home crowd which has supported me so well," Bhambri said. "You don't get to play matches against a world No. 16 every day. I'm looking forward to the next match tomorrow but hope he recovers in time for the Australian Open."
Bhambri will next take on fifth-seeded Vasek Pospisil of Canada, who was given a walkover by Lu Yen-Hsun of Taiwan, who had a right quad injury.