The French side built a 10-0 lead in the first half thanks to a penalty from Jacques-Louis Potgieter and a try on the stroke of half-time from Marvin O'Connor, which Potgieter converted.
But the second half saw the pool leaders hit back and take a share of the spoils, to remain a point in front of their French opponents in the battle for a quarter-final place.
Nicky Robinson kicked a penalty after 51 minutes and then speedy wing Christian Wade crossed for a vital try.
Wade had caused the Bayonne defence plenty of problems throughout the game with his pace and tricky running and was full value for his try.
Robinson converted and then went for a match-winning drop goal with almost the last kick of the match, but failed to hit the target.
The sides meet again at Adams Park next weekend in what will effectively be the pool decider.
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