Toby Faletau's effort had them back in the hunt at 19-13 down at half time after a slow start but they were ultimately made to pay for their indiscipline as Andrew Coombs, Lewis Evans and Rob Sidoli all saw yellow in the space of 30 minutes.
Coombs and Evans were both binned in the space of 14 minutes either side of half-time to temporarily leave the Dragons down to 13 men. Lyn Jones' side showed remarkable resilience to hold out a giant Bordeaux pack but no sooner were the captain and back-row back on the field than Sidoli was shown a yellow card and the writing was on the wall.
Jason Tovey handed the Dragons an ideal start with a penalty as early as the second minute but they were unable to kick on and instead Nicolas Sanchez took charge, slotting an equaliser before skipping through for the game's opening try, which he converted, after just seven minutes.
The Dragons had a tough time matching the brute force of the Bordeaux scrum and a string of penalties saw them trail 19-6 when skipper Coombs was shown a yellow card for tackling an opponent in mid-air. That setback spurred the Dragons back into life, though, with Wales No8 Faletau powering over five minutes before the break. Tovey landed the extras and the Dragons were well in the mix at just 19-13 sown at the interval.
But Evans' discipline slipped just two minutes into the second half when he followed Coombs to the bin, with Sanchez making him pay immediately by kicking the resulting penalty. Remarkably, the 13 Dragons withstood the flood but Robert Lilomaieva went over once Sidoli had become the third Dragon to be told to cool off.
Jefferson Poirot scored a third try late on as Bordeaux ensured the Dragons' hopes of a quarter-final place are no longer in their own hands.