O'Sullivan claims some referees have it in for him after Crucible defeat
Bingham, the 2015 champion, came through 13-10 and will face Welsh qualifier Jak Jones for a place in the final.
Jones stunned former champion Judd Trump 13-9 in his quarter-final.
"I had a little cry in my dressing room, I was fearing the worst," said Bingham. "I don't know how I've held myself together."
Bingham pulled away from an increasingly agitated O'Sullivan, reeling off the final three frames to seal a famous win.
His victory, however, was overshadowed by a bizarre incident in the 12th frame involving O'Sullivan and referee Desislava Bozhilova.
O'Sullivan was initially praised for an act of sportsmanship after seeming to play safe rather than opting to pot a red close to the black because he believed the black had not been spotted correctly.
The world number one, who was 14 points behind and on a break of 20, eventually lost the frame but was hailed for his gesture, with former world champion Neil Robertson calling it "the greatest bit of sportsmanship I've ever seen".
However, O'Sullivan hit out at officials.
"To be honest with you, some of the refs, I think they've got it in for me, so I just wanted to prove to her that she got it wrong," he told Eurosport.
"I didn't feel good about potting the ball after that, but I just wanted to make the point. The point was made. I'm not that hungry to win it in that way, so once the principle's been made I can sleep at night."
O'Sullivan subsequently told Bozhilova to "chill" after an incident in which she asked him to resume playing while he waited for some spectators to retake their seats in the arena.
Asked about the incidents afterwards, O'Sullivan tried to calm the storm.
"I won't be commenting on players or referees, it's not worth it," he added.
"I'll keep silent on that sort of stuff. I won't comment on players and I won't comment on referees. I just leave it as it is. My life's too good and I don't need the hassle."