Stevenage were level soon after but Scott Donnelly put Q.P.R. back in front before half time.
The result was put beyond doubt in the second half with two goals from Kevin Gallen.
Stevenage Borough: Julian, Fuller (Goodliffe 46), Henry (Gier 74), Santos (Oliver 46), Nutter (Wright 74), Bulman (Lewis 76), Miller (Hatton 76), Dobson (Mackie 64), Binns (Osei 80), Boyd (Nurse 55), Thorpe (Atieno 80).
Subs: Oliver, Gaia, Potter.
Goals: Dobson 21
QPR: P Jones (Cole 58), Bignot, Stewart (Rose 58), Kanyuka, Howell, Ainsworth, Oliseh (Bailey, 40), Rowlands (R Jones 58), Donnelly, Czerkas (Baidoo 58), Gallen.
Goals: Ainsworth 17, Donnelly 32, Gallen 54, 71.