Robson, a £600,000 signing in the January transfer window, gave Carl Ikeme no chance when he hit both spot-kicks with tremendous power.
Middlesbrough began the game hoping to record a double after a 5-1 win over the London side in December. Mick Harford was probably making his final appearance as QPR's caretaker manager with Crystal Palace boss Neil Warnock expected to be in charge next week.
Neither attack pressed in the early stages with the goalkeepers having little to do before Middlesbrough's Danny Coyne got down well to save from skipper Mikele Leigertwood in the 25th minute.
Middlesbrough enjoyed more possession but the visiting defence allowed them little scope in the penalty area. There was a subdued atmosphere with limited thrills for the 17,000 crowd to enthuse over.
Boro went ahead in the 39th minute when the referee ruled Peter Ramage had brought down Jeremie Aliadiere. It seemed a harsh decision when the referee pointed to the spot and QPR players were justifiably angry.
Their protests were in vain though and Robson hammered the ball past Ikeme.
Six minutes later Kaspars Gorkss brought down Leroy Lita and Robson was again bang on target.
In stoppage time, QPR were denied a goal when Coyne made a brilliant save to keep Akos Buzsaky's shot out.
QPR, not surprisingly, stepped up their pressure after the break. There was a chance for Jay Simpson to pull a goal back but he just failed to meet a left-wing cross.
Middlesbrough packed their penalty area as QPR fought to pull the game round. The home side, however, could have increased their lead in the closing stages when Gary O'Neil shot when two team-mates were in better scoring positions.