Full-back Mike Brown, released from the England training squad for the weekend, scored their first try with stand-in players Matt Hopper and Sam Smith also touching down.
Fly-half Nick Evans, back after an ankle injury, provided an instant return on the new three-year contract he signed earlier in the week by kicking 13 points for Quins.
Evans' opposite number Freddie Burns kept fourth-placed Gloucester in contention with six penalties from six attempts before converting a late try by centre Billy Twelvetrees which salvaged a losing bonus point.
Leicester had earlier moved temporarily top of the Premiership after grinding out an unspectacular 16-12 victory over their fierce local rivals Northampton at Welford Road.
Tigers could afford for young fly-half George Ford to miss five penalties and still triumph, courtesy of centre Matt Smith's second half try and 11 points by the young number 10.
On another day, Ford's inaccuracy might have proved costly, but Northampton, who were restricted to four Ryan Lamb penalties, lacked the firepower and tactical acumen to punish Leicester.
Saints led 6-3 just before half-time, but Leicester then scored 10 unanswered points - highlighted by Smith's 53rd minute touchdown - and Northampton could find no way back.
Late tries from Haydn Thomas and Chris Budgen put the seal on a convincing 33-9 victory for Exeter Chiefs against Worcester Warriors.
The Chiefs remain unbeaten at Sandy Park in the Aviva Premiership this season.
Exeter led 19-3 at half-time but the match then went off the boil, with the hosts only finding their rhythm in the final 10 minutes as they pulled further clear.
Gonzalo Camacho touched down for an early try, and Gareth Steenson booted 16 points, with Worcester's reply coming in the form of three Andy Goode penalties.