Sykes fired the home side in front after six minutes with a curling left-footed shot, having cut in from the right to finish a move started by Rob Dickie's strong run out of defence.
It was 2-0 after 23 minutes when Norwich defender Jose Cordoba was caught in possession looking to play out from the back and Joe Williams fed Wells, whose clever turn and low shot from an angle beat goalkeeper George Long.
Borja Sainz gave Norwich hope with a low, close-range finish from Callum Doyle's 82nd-minute cross, but despite late pressure they could not force an equaliser.
Liam Manning was a proud head coach after his side showed quality and grit for the win.
"It would have been nicer if we had not conceded the late goal, but the lads then put bodies on the line and I couldn't be prouder of the way they have handled a really busy program.
"They are a fantastic group. Yes, we can manage the late stages of games better, but we have a lot of young players who are still learning. They create some great moments and are prepared to fight for the team when we have backs to the wall.
"Tonight we showed some real quality at times going forward and created a lot of chances. Of course, I will look at the table and I am delighted our fans have something to get excited about."
Norwich boss Johannes Hoff Thorup hailed "positive signs" despite a second successive defeat, which sees them sit seven points adrift of the play-off places in 12th.
"It was a strange game in many ways and the reaction in the dressing room is a lot different to when we lost on Tuesday night," the Dane said.
"Then there was a lot of negativity. Tonight we know we created more chances than you can expect away from home and we lost because of very fine margins."
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