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.
There are still plenty of rough edges for Quins rugby director Conor O'Shea and his coaching staff to smooth out, yet the Irishman will be largely satisfied with a start to the season that augers well for bigger challenges that lie ahead.
Mieres booted Worcester into a second-minute lead after he was hit late by Marler, but the visitors soon found their rhythm and they dominated the opening quarter in terms of territory.
Robshaw secured a couple of early turnovers for his team, and two Evans penalties were the least Quins deserved as they pummelled away deep inside Worcester's 22.
The Warriors defence was repeatedly tested, and one technical infringement too many resulted in referee Wayne Barnes sin-binning Worcester flanker Jake Abbott.
Despite going a man down, Worcester picked themselves up and a second Mieres penalty put them on level terms, but Quins quickly snapped back into action and Care sprinted clear from 40 metres out.
Care's try was a deserved reward for the England international, who proved a constant thorn in Worcester's side with his incisive running, and Evans' conversion took Quins sevens points clear.
Worcester struggled to cope with their opponents when the visitors had ball in hand, and there was a certain inevitability about a second Warriors yellow card - this time for hooker Ed Shervington - before Marler crashed over from close range to claim Quins' second try.
Evans again converted, and Worcester trooped off 20-6 adrift after playing half the opening 40 minutes with 14 players.
Worcester had to score first in the second half to threaten any kind of a revival, but it took Quins less than three minutes before they claimed a third try, with Buchanan crossing wide out on an overlap.
Brown added a fourth Quins try after home centre Josh Matavesi received a yellow card, and although Worcester managed a late rally - Quins' replacement prop Mark Lambert was sin-binned during a spell of Warriors pressure - they ended up being comprehensively beaten.
A 68th-minute penalty try gave the Warriors a crumb of comfort from an otherwise difficult evening.
New rugby director Dean Ryan will know their discipline needs urgent attention, in addition to first up tackling that at times was not of Premiership standard.
Quins, with a livewire Care repeatedly probing openings, had far too much in the locker, while former England number eight Nick Easter - playing out of position at lock - delivered a workaholic display.
- Sports & Recreation