First-half tries from England internationals Danny Care and Joe Marler left Worcester with a mountain to climb, but they could find no way back as Quins skipper Robshaw dominated the close-quarter exchanges and repeatedly secured possession for his team.
Nick Evans chipped in with 14 points from the boot, and all Worcester could muster was two Ignacio Mieres penalties and a late penalty try that Mieres converted as their 37-13 reversal left them with just one point from three starts.
Three yellow cards during the opening 50 minutes did not help Worcester's cause, but they were outclassed by a slick Quins outfit as second-half touchdowns followed for hooker Rob Buchanan and full-back Mike Brown before substitute Ben Botica added a penalty in the dying seconds.
Quins had a bonus point safely pouched less than halfway through the second period, such was Worcester's defensive frailty, as the Londoners bounced back from last Friday's home defeat against Northampton.
They were never seriously troubled, with Worcester slumping to one of their heaviest home losses in the Premiership ahead of next weekend's Adams Park appointment with Wasps before hosting promoted Newcastle.
Quins, though, are up and running, keeping themselves in touch with the early Premiership pace-setters.
Meanwhile, Danny Cipriani came off the bench to inspire Sale to a narrow 26-22 victory over Wasps as the Sharks withstood a late fight-back.
The controversial stand-off replaced Nick Macleod after 19 minutes and racked up 21 points to ensure his side had a big enough lead to defend.
Cipriani kicked two first-half penalties and a conversion and set up Mark Cueto's try.
Andy Goode's single three-pointer was Wasps' only response before the break as they struggled for continuity.
Cipriani, who was playing his first game of the season having been left out of the previous two squads, crossed after the break to give the hosts a seemingly comfortable 20-3 advantage.
But the visitors responded well as Ashley Johnson touched down before they were awarded a penalty try.
Although Cipriani kicked another penalty, Josh Bassett's score gave Goode the chance to put them ahead with the conversion, but the fly-half missed and Sale's mercurial pivot sealed the win with a last-minute three-pointer.