World number 70 Advani, who beat Ruhr Open winner Mark Allen in the second round, took another huge scalp in the last-16 as he earned a tense win over the world number 12 with a superb 86 in the final frame
Advani had raced into a 3-1 lead despite Bingham firing a 117 in the second frame.
Bingham struck back with breaks of 72 and 119 to force a decider and got a start in the seventh frame as he got to 30 points, but let his opponent in and Advani grabbed his chance brilliantly.
Mark Williams was also shocked by a home hope in round three going down 4-2 to Aditya Mehta.
Williams took the first frame before breaks of 54 and 66 secured Mehta an advantage that Williams immediately cut in a scrappy fourth frame.
However, Mehta, seeded 71st for the tournament, took the next two frames to move into the next round.
Ding Junhui also made it into the quarter-finals with a 4-2 win over John Higgins, while Anthony McGill and Robbie Williams also came through.
Neil Robertson, Michael White and Stephen Maguire also progressed to quarter-final stage.
Indian Open last-16 results:
DING JUNHUI 4 v 2 JOHN HIGGINS
ANTHONY MCGILL 4 v 2 JOE PERRY
MIKE DUNN 0 v 4 ROBBIE WILLIAMS
STUART BINGHAM 3 v 4 PANKAJ ADVANI