A penalty from Gordon Ross was Welsh's only response, and a Welsh mistake opened the doors for Monye to score his second and put Quins 21-3 up at half time.
Mike Brown made a break 15 minutes into the second half to set up George Robson for his second of the night, with the youngster diving over from close range.
Brown was at the centre of things again, as a chip over the Welsh defence set George Lowe free to score Harlequins fifth try of the night. Once again, Nick Evans kicked the extras.
Ill discipline saw Welsh reduced to 14 men in the final minutes and Harlequins took full advantage, with replacement Ben Botica landing the final blow.