After defeat to Bath last time out, Saracens were sluggish early on at London Stadium as Quins opened up a 17-3 advantage, Marcus Smith scoring 12 points.
Two penalties from Alex Lozowski were all Saracens had to show at the break but the defending champions rallied with Will Skelton grabbing two tries.
Schalk Burger began Sarries' comeback as he barged over from close range, before Skelton turned the game around in the space of nine minutes.
It was a second successive defeat for Quins, who drop to fourth in the table after Gloucester beat Wasps 27-14 at Kingsholm.
After a sluggish start, Gloucester eventually found its rhythm, fighting back from 11-0 down to lead 14-11 at the break.
Wasps were unable to replicate their impressive start after the break as Jake Polledri's try and three penalties from Billy Twelvetrees gave Gloucester victory.
Elsewhere, Worcester Warriors boosted their survival hopes with a 27-25 win over Bristol Bears, while fellow strugglers Newcastle Falcons made it three wins in-a-row with a 22-17 triumph over Sale Sharks at St James' Park.