Bothroyd, inset, back from a loan spell at Sheffield Wednesday, settled the tie when he headed home a corner from Alejandro Faurlin.
Bothroyd was a constant threat but Rangers were indebted to several fine saves from goalkeeper Robert Green. On 29 minutes Romanu Lukaku got on the end of Goran Popov’s probing centre and his glancing header was turned away at full stretch by Green.
The second half saw little in the way of clear-cut chances until Bothroyd’s late winner. With 18 minutes left, the former England man 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.
* QPR manager Harry Redknapp last night said he expects to complete the signing of Marseille striker Loic Remy in the next 24 hours and hopes to sign Rennes midfielder Yann M’Vila in the next few days.
WEST BROM: Myhill, Jones, McAuley, Tamas (El Ghanassy 81), Popov, Dorrans, Morrison (Reid 90), Thorne, Thomas (Odemwingie 71), Lukaku, Rosenberg.
QPR: Green, Onuoha, Hill, Ferdinand, Ben Haim, Mackie (Taarabt 90), Park, Derry, Faurlin, Bothroyd (Mbia 80), Cisse (Wright-Phillips 46).
Referee: Martin Atkinson.