The Dragons rallied and had a great deal of possession but could not convert it into points, with replacement flyhalf Dan Griffiths guilty of missing three penalties after coming on for Nathan Brew.
And though the Dragons had plenty of ball they were constantly pushed back by the Ospreys' offensive defence.
Both sides were reduced to 14 men before halftime when Andrew Millward and Rhys Oakley were sent to the sin bin for technical offences, but the Ospreys still went into the break 10-0.
Skipper Barry Williams put scrum-half Jason Spice in for the next Ospreys' score soon after the half-time break and their lead was stretched further when Connor added a penalty moments later.
Another try from Spice put the result beyond doubt, and they secured the bonus point when Adrian Durston crossed in injury time.
But Lyn Jones will have to wait at least one ore week to wear his Celtic crown as Munster registered victory over Ulster to keep their slim hopes alive.