He struck in the 15th and 88th minutes, while Gavin Rae levelled for an under-par Cardiff on the stroke of half-time.
All the Bluebirds came through the bruising encounter at Stoke unblemished so they were unchanged for the visit of Norwich, who were on an 11-match unbeaten run.
In the opening 30 seconds, Cardiff won a corner but in clearing the danger, Mo Camara was injured and the Canaries cleared their lines with a free-kick.
The visitors went ahead when the home defence failed to clear the ball after a run down the left flank by Ryan Bertrand.
He laid it back to Evans, who was waiting on the edge of the penalty area, and the youngster rifled a low shot into the far corner.
Only a sliding tackle by Glenn Loovens midway through the half prevented Norwich from increasing their lead as their quick breaks out of defence continued to give the Bluebirds problems.
Jimmy Floyd Hasselbaink worked a rare opening in the 37th minute but he was off target with his shot, while then two minutes later he turned sharply inside the box but again dragged his shot wide.
City equalised on the stroke of half-time when Matthew Bates was caught in possession by Joe Ledley, who took the ball towards goal before slipping a pass inside to Hasselbaink.
The veteran striker fed Rae and the former Rangers midfielder struck his shot beyond two defenders and into the net to level the scores slightly against the run of play.
The Bluebirds were quicker to the ball at the start of the second half and the pinned the Canaries back for long periods, but were unable to prise out any openings.
A superb cross from Tony Capaldi was met with a Paul Parry header, but Canaries keeper David Marshall flung himself across to turn it round the post.
But Evans struck a dramatic winner for the Canaries when he gathered the ball just inside the Cardiff half before unleashing a tremendous shot from 35 yards that was good enough to win any game.
There was little time for the Bluebirds to regroup and their five-match winning home run came to a disappointing end.