Worcester dominated possession and territory but their opponents were more clinical as the Warriors wasted opportunities and could not fully capitalise on the sin-binning of both Quins centres Jamie Roberts and George Lowe.
The Warriors' first-half performance was probably their worst of the season and Quins did not have to do anything exceptional to lead 24-3 but they were transformed in the second period and in the end were unlucky losers.
Charlie Walker (two) and Luke Wallace scored the tries for Quins, with Nick Evans converting all three and adding a penalty.
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Worcester responded with tries from Marco Mama and Phil Dowson and a penalty try award which Ryan Mills converted and Tom Heathcote kicked a first-half penalty.
Quins made a dream start, taking the lead within three minutes from their first attack. The visitors moved the ball wide from a line-out near the halfway line before Walker and Ross Chisholm combined well to give Walker the opening try, which Evans converted.
Worcester responded and Heathcote missed a penalty after David Ward had obstructed Chris Pennell. He had more luck with his second attempt to make it 7-3 after Lowe had been penalised and sin-binned for a deliberate offside.
Lowe returned to the action with no damage to the scoreboard and the visitors went further ahead in the 28th minute. They declined a kick at goal in favour of an attacking line-out and were rewarded when Wallace forced his way over for the try, again converted by Evans.
There was controversy two minutes later when Heathcote was pole-axed by a challenge from Quins' giant lock Charlie Matthews. Referee Wayne Barnes sought the advice of his television match official but the challenge was deemed fair with Heathcote leaving the field to be replaced by Andy Symons.
The situation became even worse for the hosts when Evans fired over a penalty before Worcester had a chance to reduce the arrears. With Heathcote absent, they declined two kickable penalties in favour of line-out drives but a knock-on ruined their chances and they were firmly up against it as they went into the interval 17-3 behind.
Quins looked to have sealed the game within four minutes of the restart when Walker collected a clever chip through from Danny Care for the try, which Evans converted to put the visitors seemingly out of sight.
Worcester rallied for Mama to crash over from a driving line-out before Roberts was yellow-carded and immediately after that, Barnes awarded the hosts a penalty try for a scrum infringement which Mills converted.
Worcester's second half dominance was rewarded further with 10 minutes to go when Dowson scored Warriors' third try despite Walker seemingly being impeded as he sought to retrieve the loose ball. Dowson seized on it to score.
Dowson's try set up a grandstand finish but Roberts' return enabled Quins to stem the tide and just hold on.
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