Gloucester had looked set to lose their 100 per cent record as they trailed 26-25 heading into the final stages, but Burns capped a fine individual night with the boot with a brace of three-pointers in the 75th and 80th minutes that earned the Aviva Premiership side victory.
England international Burns, 22, finished with 26 points after notching eight penalties in total as well as converting Darren Dawiduik's try on the half-hour.
Bordeaux, who had led for large parts of the match, scored two first-half tries through Rafael Carballo and Tamato Leupolu but they could not hold on as Burns snatched the win for the visitors.
Gloucester, the 2006 tournament champions, now have five wins from five in Pool One and a berth in the last eight.
French club Bordeaux, meanwhile, remain bottom of the group with two points after suffering a fifth successive defeat.
- Sports & Recreation