Castres have conceded 206 points in five straight away defeats in the Top 14 this term, including a 59-7 thrashing in Bordeaux last weekend.
Harlequins, meanwhile, had crushed London Welsh 52-0 in their most recent home match, so it was easy to see why most pundits were predicting a high-scoring success for the Premiership outfit.
However, the French side belied those stats to stay in touch for over an hour before Care celebrated the birth of his new baby with the only try of the game.
Evans set up that score and contributed 20 points of his own with six penalties and a conversion in a perfect kicking display.
The visitors started brightly and were rewarded for some early invention when awarded a penalty which Rory Kockott stroked over, but Evans levelled the scores on 23 minutes when Castres were pulled up for slowing things down at the breakdown.
The home side seemed to be warming to the task but could not capitalise on good field position following breaks by Luke Wallace and Nick Easter and half-time arrived with the score at 3-3.
Quins edged ahead for the first time when Evans' aim was true with a 44th-minute penalty from just to the right of the posts, but Kockott replied in kind with an almost identical strike for 6-6.
Evans was successful with two more penalties before the number 10 produced a moment of magic to settle the tie when chipping over the Castres defence, re-gathering the ball and then kicking on again for Care to race through and collect before crossing.
Kockott pulled three points back with a 69th-minute kick but the Quins forwards dominated the closing stages and Evans slotted over two more penalties to wrap things up.