The home side ran in five tries which were scored by Matthew Pewtner and Hallam Amos, Patrick Leach, Toby Faletau and Ashley Smith at Rodney Parade.
Fly-half Jason Tovey added to the points by kicking a personal haul of 15 points with the boot.
Prop Jefferson Poirot and Jean Baptiste Mamotte crossed for tries for Bordeaux, who were also awarded a penalty try.
But the Men of Gwent, who welcomed back three of their international players, secured a vital victory to keep the pressure on unbeaten Bath.
Bordeaux-Begles, who are currently 10th in France's Top 14, will host the regional side at home next weekend.
Dragons Team
Daniel Evans, Matthew Pewtner, Pat Leach, Ashley Smith, Hallam Amos, Jason Tovey, Wayne Evans; Hugh Gustafson, Sam Parry, Dan Way, Andrew Coombs (c), Matthew Screech, Lewis Evans, Nic Cudd, Toby Faletau.
Replacements: Elliot Dee, Owen Evans, Nathan Buck, Rob Sidoli, Ieuan Jones, Richie Rees, Kris Burton, Ross Wardle.
Bordeaux-Begles Team
Joaquim Tuculet, Robert Lilomaiava, Charles Brousse, Thibault Lacroix, Rafael Carballo, Romain Lonca, Manu Saubusse, Jefferson Poirot, Benat Auzqui, Benjamin Sa, Cameron Treloar, Aliki Fakate, Gauthier Gibouin (c), Matthew Graham, Poutasi Luafutu.
Replacements: Jean-Charles Fidinde, Laurent Del Boulbes, Patrick Toetu, Adam Jaulhac, Jean-Blaise Lespinasse, Baptiste Serin, Gilem Queheille, Jean-Baptiste Lamotte.
