Darren Edwards’ men appeared on course for a famous victory over the Basque club after leading by 13-points mid-way through the second-half.
But following a quick-tap the Wales scrum-half bulldozed through three of his fellow countrymen to snatch the match-winning try.
Jacques-Louis Potgieter opened the scoring at Stade Jean Dauger and he was on target once again after Prydie opened the Dragons’ account.
But the Men of Gwent instantly hit-back with Michael Poole charging down Potgieter’s clearence and Lewis Evans pounced for the try.
Jones added the conversion and Prydie stretched the Dragons lead to 10-points with two more penalties before the interval.
Prydie got the scoreboard ticking once again following the break but Bayonne called on the big-guns and they made an immediate impact.
Davied Roumieu benefitted from a monstrous driving-mal and Potgieter added the extras.
The Dragons battled to hold onto their lead but Bayonne were relentless. Phillips came close following a razor-sharp attack but the ball was recycled and spread wide for Sam Gerber to cross the whitewash.
Bayonne failed to add the conversion and Prydie responded with a penalty to stretch the Welsh region’s lead back to four points.
But they were unable to hang on as Phillips rampaged over to break Gwent hearts.
