Saracens prop Ben Broser scored the only try of this low-scoring match, in the 14th minute.
But not even the bonus point they earned for losing by less than seven points could improve their position. Saracens are still second from bottom and have not been able to secure two Premiership victories in succession this season.
Burke, the former Ireland international, slotted the first of his four penalties in the third minute, before Broser went over for his try in the 14th minute. Robbie Kydd added the conversion for a 7-3 lead to Saracens.
Quins continued to hammer away at Saracens and penalties in the 16th, 34th and 40th minutes saw them take a 12-7 lead into the break. The third penalty landed by Burke was from well inside his own half.
Burke, who did not fail with a kick up till then, was stretchered off nine minutes into the second half with a neck injury, after a collision with England flanker Richard Hill.
The score remained unchanged till the 73rd minute, when Kydd slotted a penalty, but that was all too little too late for Saracens.
For NEC Harlequins:Pens: Burke 4
For Saracens:Try: BroserCon: KyddPen: Kydd