Robinson was suspended last week following Bristol's poor start to the season
Robinson was suspended last week following Bristol's poor start to the season and chairman Chris Booy confirmed on Monday he wouldn't be returning because the club didn't believe he could halt the team's poor run.
After Bristol failed to win any of their seven league games this campaign, Mark Tainton was appointed as acting head coach.
Robinson had been in charge since March 2013 and led the club to the Championship title and promotion to the Premiership last season.
"We thank Andy for his contribution to the overall coaching set up at the club. Unfortunately, the board lost confidence that Andy would retain the club's Premiership position, bearing in mind the results to date," Booy said.
"Andy departs with the knowledge that he made a significant contribution in helping Bristol return to the Premiership this season."
Robinson signed a new three-year deal with Bristol in August after helping the west country outfit end their seven-year exile from the top flight.
But the club sit seven points adrift at the foot of the Premiership table with just two points from seven games.
They were also beaten by Cardiff Blues and Bath in the European Challenge Cup and went down to Worcester on Friday in their opening Anglo-Welsh Cup clash.
"As everyone appreciates, the start of the season for Bristol Rugby has been extremely tough. However, I was surprised and disappointed by the board's decision," Robinson said.
"I would like to thank everyone involved with the club for some very special memories and wish them all the best for the future."