James Gray and Kal Naismith scored the goals for James Beattie's side, who had to come from behind to triumph after Alex Henshall put the Pirates ahead.
Rovers made the dream start when home goalkeeper Marcus Bettinelli fumbled Henshall's left-wing cross into his own net to give John Ward's side a fifth-minute lead.
But that lead lasted just 14 minutes with Gray levelling from close range from Peter Murphy's fine pass.
And that goal jolted the Reds into life, with Naismith drilling a low shot into the left corner of the net to put Beattie's boys in front just shy of the half hour mark.
That led to a spell of pressure for the home side with Danny Webber denied by Steve Mildenhall's near post save.
A cagier second half brought fewer chances, with Rovers keeper Mildenhall turning a Lee Naylor free-kick over the crossbar before substitute Eliot Richards hit the Stanley bar for the Gas.
Accrington held on for their first three-point haul but they remain bottom of Sky Bet League Two.