Winger Monye scored after just eight minutes, Gonzalo Tiesi crossed three minutes later and Mike Brown claimed a 13th-minute touchdown to stun the home fans.
Matthew Jones hauled Worcester back towards respectability, converting his own try after kicking a drop-goal and a penalty.
But fly-half Chris Malone carved out a second try for Monye (right) before the break — Quins’ bonus point score — and it was damage-limitation for Worcester after that as they changed ends 27-13 adrift.
Chris Latham set up Warriors wing Miles Benjamin for a try early in the early period.
But Malone’s second penalty restored a 10-point cushion before Jones landed three late points.
The only cloud for Quins was losing England winger David Strettle with a foot injury at half-time.
Boss Dean Richards said: “David came off as a precaution and we will have to wait for the results of a scan.”
Worcester coach Clive Griffiths said: “It is is not happening for us at the moment.
“We ‘won’ the second-half but we will look back on this game as an opportunity lost.”