The pair were neck and neck at 10-10 going into the interval but Robertson looked to have taken control of the match when winning a black-ball decider in the next frame to edge ahead.
However Fu kept his cool to take the next three frames and set up a quarter-final showdown with defending champion Selby, who earlier in the day had comfortably beaten Xiao Guodong 13-6.
Fu is now bidding to reach his third Crucible semi-final and was just relieved to get through against Robertson following a scrappy final few frames.
"It was very tough. I had so many chances and missed so many chances,” he said. "It was one of those matches that neither of us deserved to win. For fighting spirit I was a 10 out of 10, but for snooker it was four out of 10."
Meanwhile, Robertson was far from happy after the match and described his performance as "garbage".
"I played awful snooker," said the Australian. "It wasn't good to watch. I was awful in my first match too."