The fifth seed started a set down after Monday's downpour but recovered to record a 4-6, 7-6 (7/4), 6-4, 6-4 success, despite two more lengthy rain delays.
Tsonga, a beaten semi-finalist last year, celebrated with his familiar dance and will face Philipp Kohlschreiber in the last eight after the 27th-seed beat American qualifier Brian Baker 6-1, 7-6 (7/4), 6-3 on Court 12.
Tsonga expects Kohlschreiber to raise his game considerably in his first SW19 quarter-final.
The 27-year-old Frenchman said: "He has never reached the semi-finals, so of course he will play hard. I expect to have a very, very tough match.
"Maybe you will say I'm the favourite because I have played against him now many times. So I'm the favourite. I just have to play my game."
Fish was left to rue what might have been after he watched his overnight lead dissolve amid the rain delays on Court Two.
He said: "I went to bed last night and I thought I could maybe win the tournament the way I was playing.
"I played well enough to win. I would have beaten a lot of players. I was certainly satisfied with the way I competed and the way I played.
"He did not play as well yesterday as he did today. I played better yesterday than I did today maybe.
"So who knows what happens if there's not a rain delay."
Florian Mayer finished off Frenchman Richard Gasquet to line-up a Court One quarter- final against top seed Novak Djokovic, who beat fellow Serb Victor Troicki on Monday.
Mayer claimed the first two sets before the Frenchman hit back in the third, however, the German held on to win 6-3, 6-1, 3-6 6-2.
Meanwhile, the Swiss six-time champion Roger Federer and first-time quarter-finalist Mikhail Youzhny, who both won through on Monday, open the play on Centre Court tomorrow afternoon at 1pm.