The substitute converted from the spot after winning the penalty herself, after she was introduced as a substitute in a game that saw few chances for either side.
The first sight of goal for the Cherries went the way of Elena Cole, but her shot from distance was comfortable for QPR keeper Kenny Clarke.
Bournemouth were restricted to efforts from range early on, as Chloe Newton was next to try her luck, but her shot was off target.
Katie James headed over the crossbar after a Cole cross, before she saw a shot deflected wide.
Bournemouth awarded a penalty in the first half but James saw her spot kick saved by Clarke to her right to keep the game level at 0-0 with two minutes left of the first half.
The second half saw neither side threaten in the early exchanges, and it was substitutions from Helen Bleazard which saw the tide shift in favour of the Cherries.
Sophie Quirk and Thompson were both introduced with 19 minutes to play, and both applied pressure on the QPR backline.
Quirk had some joy against her marker and a cross that was caught by Clarke was a signal of intent from the forward.
She again beat her marker, cutting back to fellow sub Thompson who saw her shot blocked in the box.
The Cherries were awarded another penalty when Thompson was fouled in the box after picking up a loose ball as QPR failed to clear Jenna Markham’s cross after she was slipped in behind by Quirk.
And the Cherries number seven showed composure from the spot to place her penalty into the bottom corner and hand Bournemouth a late lead.
Kelci Bowers went close with a header but she was denied by a fantastic Clarke save down to her left.
Quirk was then also denied by Clarke when one-on-one as the Cherries sought to make their advantage more comfortable, but QPR were unable to draw themselves level and Bournemouth booked their spot in the second round.
The draw will take place tonight at 7.15pm, live on the FAWNL Instagram channel.