And while both sides struggled to gain control of the match in testing second-half conditions, Shillcock made the breakthrough late on to hand the visitors the win.
The result sees Worcester win on the road for just the second time in 14 months, while Quins' stuttering start to the campaign continues with their first home defeat since April.
After a few cagey opening minutes – in which the two sides exchanged a number of kicks – visiting skipper Lance got the game's first points from the kicking tee following successive penalties in Worcester's favour.
But it did not take long for Harlequins to respond. Marcus Smith knocked a penalty deep into Worcester territory and following good pressure from Francis Saili, Dombrandt slipped through a gap in the away defence to go over under the posts with less than 10 minutes gone.
Saili was at it again moments later, tackling Shillcock under pressure in his own 22 to set up a loose ball that Chisholm capitalised on, helping Quins to a 14-3 advantage with just under quarter-of-an-hour on the clock.
The Warriors exerted a greater influence on the game in the second quarter and had the Quins backline pegged back to their five-metre line for a number of phases, but steely home defence did enough to keep them at bay.
A second Lance penalty reduced Worcester's arrears in the 33rd minute, and after an even period of play, the Australian kicked three more points with the last action of the first-half to make it 14-9 at the break.
The visitors started the second half in determined spirits and a testing grubber kick from Ed Fidow put them within inches of the home tryline, but once again Harlequins stood firm to retain their lead.
In miserable conditions, both teams struggled with the ball in-hand and Chisholm and returning Fiji international Vereniki Goneva were both guilty of spilling the ball in strong positions.
But the Warriors rallied and following a sustained period of pressure, Shillcock bulldozed over in the corner, before Lance added the extras to give them the lead and set up an enticing last 12 minutes.
And a last-gasp Lance penalty added to the scoreline in the closing stages, as Worcester held on for a crucial victory ahead of two rounds of European action.