The initial tie at Loftus Road ended in a 1-1 draw earlier this month but Harry Redknapp's men were able to edge a win on Tuesday night when Bothroyd, back from a loan spell at Sheffield Wednesday, climbed to nod home Alejandro Faurlin's corner in the 76th minute.
The result sees QPR, currently bottom of the Premier League, advance to the fourth round where they will host MK Dons, while it leaves Steve Clarke's side with only two wins from 11 games in all competitions.
The Hoops, showing six changes from the side which drew with Tottenham at the weekend, were the first to threaten with Bothroyd having a shot deflected wide.
West Brom's on-loan striker Romelu Lukaku then teed up an early opportunity for the overlapping Jerome Thomas inside the box, but the winger surprisingly elected not to shoot when in a dangerous position and his cross was easily dealt with.
Visiting defender Tal Ben Haim became the first player to be booked after 18 minutes for bringing down Lukaku.
Robert Green, who enjoyed an excellent night between the sticks, made a fine save to keep his side on level terms after 29 minutes. Lukaku got on the end of Goran Popov's probing centre and his glancing header was turned away at full stretch by Green to his left.
The game started to come to life and Boaz Myhill parried away Bothroyd's 25-yard drive. Thomas was seeing plenty of the ball and another surge into the QPR box forced Green to parry away his cross.
Redknapp brought on Shaun-Wright Phillips for the ineffective Djibril Cisse at the start of the second half, while Thomas was yellow-carded after 51 minutes for a challenge on Jamie Mackie.
QPR were showing more purpose and Gabriel Tamas blocked a powerful attempt from Faurlin. Myhill then had to adjust to keep out a deflected drive from Wright-Phillips and Green also reacted sharply to cling onto Lukaku's low shot after his run to the edge of the QPR area.
Home striker Peter Odemwingie, a transfer target for Redknapp, replaced Thomas with 18 minutes left, but it was Bothroyd who broke the deadlock three minutes later. He got plenty of power into his downward header from Faurlin's corner and Myhill could only help the ball into the corner of the net.
Albion tried to rescue the tie and Green again excelled in twice beating out powerful shots from the previously ineffective Markus Rosenberg. With time running out, the keeper also denied Dorrans as West Brom could not force the game into extra-time.