There were few chances of note in the opening exchanges, with Cameron Jerome coming closest from a corner, but he couldn't keep his header on target.
Fulham went in front when Kavanagh drilled the ball low and beyond John Ruddy after being allowed too much space on the left flank.
The goal gave the home side confidence and they didn't allow Norwich to have any time on the ball as the half went on.
A dramatic start to the second half saw Jerome go down in the box for a Norwich penalty, but Lewis Grabban slammed the spot kick against the bar to waste a great opportunity to get his side back in the game.
The missed chance spurred Fulham into action, with Lasse Vigen Christensen and McCormack putting pressure on the Norwich goal.
McCormack swung a free kick towards Rodallega but Bradley Johnson was there to nick the ball away.
Johnson then fired a shot from range that had Marcus Bettinelli scampering across his goal before substitute Josh Murphy hit the post, but Norwich could not find the equaliser and so Fulham secured a 1-0 win.