The Ospreys had lost the last three meetings between the sides but tries courtesy of Ross Jones, Hanno Dirksen, Eli Walker and South African centre Stefan Watermeyer gave them their first win in this season's Anglo-Welsh competition, with fly-half Matthew Morgan kicking three conversions.
The Dragons had been level at 14-all at the interval thanks to a try from wing Pat Leach and three penalties from fly-half Steffan Jones, but replacement Luke Williams' last-gasp try was their only score of the second half.
Morgan missed an early penalty but the Ospreys made an ideal start by scoring the opening try inside four minutes. Wing Hanno Dirksen made the break, after Dragons wing Leach had coughed up possession, to allow Ross Jones to step the cover and dot down, with Morgan adding the extras.
Jones replied with a penalty for the Dragons before some typically direct running from centre Andy Tuilagi carved the Ospreys defence open, only for the Samoan's final pass to find home winger Walker.
Another Jones penalty trimmed the gap to a point but the hosts responded with a well-worked try.
The Dragons missed their man at the line-out and smart work from scrum-half Tom Habberfield and centre Watermeyer freed up the space for captain Sonny Parker to send Dirksen scampering over.
The lively, open start continued as the visitors hit back with their first try. Fluent passing along the backline allowed Leach to skate down the flank and see off a high tackle from the covering Dirksen to flop over.
Jones struck the post with the conversion, but added a monster penalty to tie the scores at 14-all, minutes after Morgan had pushed a straightforward attempt.
The Dragons were doing themselves few favours in the discipline department as they conceded a regular stream of penalties throughout the opening half, although the off-colour Morgan let them off by missing touch from several infringements.
Further ill-discipline gave the Ospreys a series of five-metre scrums towards the end of the opening 40 minutes, but the Dragons toughed the period out.
Morgan missed a third penalty effort early in the second half but the Dragons defence was breached again in the 48th minute as controlled build-up laid the groundwork for Walker to wriggle out of a tackle to go over to the right of the posts, Morgan making no mistake with the extras for a 21-14 lead.
The bonus-point try was not long in coming, with a lovely pass from flanker Sam Lewis sending Watermeyer clear for a try on his Ospreys debut, although Morgan could not add the touchline conversion.
The Dragons should have hit back when replacement Geraint O'Driscoll's wide pass gave Tuilagi plenty of room in the home 22.
The slight frame of Morgan clung on bravely to make a try-saving tackle, although there was a cost for the Ospreys as Canada flanker Chauncey O'Toole was binned after coming in from the side at the subsequent ruck.
Despite the numerical disadvantage the Ospreys continued to threaten in attack.
Morgan missed a simple drop goal shot before replacement scrum-half Tom Grabham's smart break launched a promising attack from deep, forcing O'Driscoll into desperate clear-up work on his own line.
The Dragons had the last word as replacement Williams finished off a good backline move at the death, before adding the conversion.