Hong Kong star Fu led 2-1 after hitting a break of 110 before O'Sullivan split the tip of his cue and needed to make running repairs at the interval which was taken early.
The duo shared the next four frames, as the Far East star kept his nose in front, but the Englishman then reeled off three frames in a row, with a trio of half-century breaks, to take the match 6-4 and progress to Sunday's showpiece.
The five-time world champion, who will play either Barry Hawkins or Joe Perry, felt it was his top display in north London under trying conditions.
"It is probably the best match I have won, given the circumstances," he told BBC Sport. "The tip was gone, completely gone. It just couldn't take any chalk. I mis-cued five or six times. It was like chalking a bit of slate.
"I was going to wait for the interval but it was so gone and they said 'look, you can take the interval now' and that was sweet."