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.
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
$29.99
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.
