It was only the Dragons fifth' win in their last 13 fixtures in the competition but it sets them up nicely for a daunting trip to France to face the same opposition next Thursday.
It took the Dragons just seven minutes to get the game's opening try when Wayne Evans and Jason Tovey combined twice in midfield put centre Leech over in the corner. Fly-half Tovey added the conversion and the Rodney Parade crowd had barely sat back in their seats when the Dragons got their second.
Scrum-half Evans burst clear down the touchline before finding Faletau on his inside shoulder and the Wales star drew Argentine full back Joaquin Tuculet and sent wing Pewtner clear to score.
Bordeaux No9 Manu Saubusse missed the chance to claw some points back with a penalty on 12 minutes but made up for it 10 minutes later with an easier kick from close range.
The Dragons survived a difficult 20 minutes with just three points conceded and it wasn't long before the home side grabbed their third try.
Youngster Amos picked a great line to take a short ball off Evans and burst through the gap to score, with Tovey adding the extras.
Bordeaux hit back before the break with a try from prop Jefferson Poirot but a Tovey penalty just before and five minutes after half time extended their lead to 15.
The visitors refused to lay down and were awarded a penalty try on 55 minutes, but their comeback lasted just two minutes as Faletau raced in for the Dragon's fourth try.
Amos had a hand in the Dragons' final try as he created space for Ashley Smith to cross seven minutes from time.
A late try from replacement Jean-Baptiste Lamotte meant the Dragons had to do some defending to prevent the visitors getting a losing bonus point but they held out well to pile the pressure on Pool 2 pace setters Bath.