There was little to separate the two sides in the opening 40 minutes, as Morne Steyn added two penalties for the home side, while Nick Evans kicked three for the Aviva Premiership side.
The visitors increased their lead early in the second half when Mike Brown found a gap with a move inside to score the opening try of the night.
Quins were able to frustrate the hosts in the scrum and lineout to make sure that they did not have a chance to get back into the tie, while Evans extended their advantage to 16 points with a drop goal.
Tim Molenaar was able to round off the emphatic win with a try in the corner inside the final five minutes, after Nick Easter passed the ball out wide to him.
Harlequins will face Northampton Saints in the last four later this month.