A pre-match downpour in south-west London reduced the game to a battle up front and Evans' 11 points proved the difference between the sides.
On a day when relegation rivals Newcastle Falcons bagged an all-important win, Richard Hill's Warriors played their part in gaining a losing bonus point - Aviva Premiership fly-half debutant Joe Gray kicking nine points to add to a Miles Benjamin score.
A first-half penalty try was Quins' other score but it was Evans' boot - despite two second-half misses - that sealed a hard-fought win.
Worcester were rewarded for some incessant early pressure when Gray kicked a sixth-minute penalty.
The Warriors were eight points to the good a minute later after Benjamin, having very nearly scored in full flow seconds before, dived over from close range.
George Lowe, returning for his first Harlequins start since November, produced a try-saving tackle to seemingly deny the flying Worcester wing, however Benjamin bundled over seconds later.
Gray put his conversion attempt from out wide just past the left upright and that miss seemed to spring the home side into life.
Full-back Mike Brown, returning to domestic action from the England set-up, went agonisingly close to opening Harlequins' account but could not get his hands on a chip ahead.
Just before the midway point in the first half the free-flowing visitors went close to stretching their advantage.
Winger Marcel Garvey was the man making the 60-yard break
but he was unable to show the required composure when the tackles arrived, a wayward pass failing to put Benjamin clear.
Quins finally got themselves on the scoreboard after 22 minutes through an Evans penalty before number eight Nick Easter threatened a try.
Easter was eventually stopped in his tracks, partially by a Garvey tackle. The short-term gain from the outgoing winger was severely tempered by a nasty-looking mouth injury sustained in making the hit.
Garvey's departure galvanised Quins and they were rewarded for the dominance of their pack with a 29th-minute penalty score.
Quins built up ahead of steam from the 22 and, when the try line was in site, Worcester hooker Ed Shervington was sin-binned for pulling down the maul.
Easter was the man denied the chance to cross but referee David Rose awarded the penalty try and Evans easily added the extras to put Quins ahead for the first time.
The home side's domination up front continued after the half-time break, and when Worcester were penalised at scrum time after 43 minutes Evans' penalty stretched the home side's lead to 13-8.
Worcester hit back just minutes later when their third-choice fly-half Gray slotted a penalty to reduce the deficit to just two points.
Quins again gained a five-point advantage just after the hour mark with another Evans three-pointer, but the Warriors pack refused to lie down and, after winning a penalty from a scrum, Gray made it 16-14.
Evans missed two kicks at goals as the dominant Quins scrum again earned two late penalties but they proved academic.
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