The Dragons led 20-13 at the break in the Pool Three clash at Rodney Parade, but a second-half rally earned Newcastle a 30-26 victory.
Newcastle were staring down the barrel as Matthew Pewtner, Taulupe Faletau and Ashley Smith all ran in first-half tries for the Welsh side.
However, John Wells' men were kept in the contest by the boot of Rory Clegg and they capitalised on soft defending by the Dragons, as Mark Wilson added to Lee Smith's first-half try before Will Witty took advantage of a mix-up between Pewtner and Hallam Amos.
Newcastle are now top of the pool with nine points, but a losing bonus point for the Dragons leaves them just three points adrift in second.
Edinburgh also have two wins thanks to a 25-17 triumph over Lyon in Pool Four, despite a spirited second-half showing from the away side.
London Irish picked up another bonus-point win in Pool One as scores from Fergus Mulchrone, Halani Aulika, Alex Lewington and Luke Narraway earned a 25-15 success against Grenoble and ended the hosts five-match winning streak at the Stade Lesdiguieres.
Elsewhere, La Rochelle bounced back from last week's 48-12 loss to Connacht by beating Bayonne 25-13 in Pool Two, while Brive remain winless after they fell to a narrow 26-21 loss at home to Zebre in Pool Five.