Christiana Balogun barged over from close range to put the Bears ahead, but Beth Wilcock skipped past two tackles to score and edge Quins into the lead, before Aoife Wafer extended their advantage to 14-5.
Bristol wasted opportunities before the break as Ella Lovibond’s pass was too long for wing Millie David, who would have had a walk-in to the line, and Delaney Burns was then held up under the posts.
Yet Orr turned the game on its head for Bristol midway through the second half, collecting and touching down a chip kick from centre Meryl Smith to cut the gap to two points.
Ten minutes later Orr doubled her tally with a standout solo try, shoving off one tackler and side-stepping two more to run over and make it 17-14.
At the other end replacement Quins prop Jessie Spurrier was held up as the London side were unable to put any points on the board during the second half.