The contest started in lively fashion, with both sides showing good attacking intent despite a lack of chances in the opening minutes.
Leighton Baines seemed to carry the biggest threat for the visitors with his ventures up the left hand side of the pitch but good work from the England international was ruined by a poor shot from Steven Pienaar.
Fellaini had the opportunity to give his side the lead when he connected with a Baines corner but the Belgian failed to test John Ruddy in the Norwich goal.
Kevin Mirallas came close to scoring when he dragged a shot wide of the target following a flowing passing move from the visitors.
Half-time came at a good stage for Norwich, who were beginning to lose control of the match with Everton threatening with almost every attack.