Everton striker Duncan Ferguson scored twice from the penalty spot to send Norwich crashing out of the FA Cup.
The home side took the lead thanks to Kevin Kilbane's close-range header from Thomas Gravesen's free-kick.
But Norwich midfielder Jim Brennan equalised midway through the first half with a brilliant left-foot volley.
However, Norwich striker Iwan Roberts conceded two penalties in each half with Ferguson converting both.
15 mins: Kilbane heads Everton into the lead
21 mins: Brennan equalises for Norwich with a well-taken shot
38 mins: Ferguson scores from the penalty spot
70 mins: Ferguson converts second spot-kick
Everton dominated the opening period, testing Norwich's nervy defence with a barrage of aerial crosses for target man Ferguson.
The Everton striker forced a superb save from Norwich goalkeeper Robert Green, who tipped over Ferguson's powerful header from Kevin Kilbane's pinpoint cross.
Gravesen was in the thick of the action for Everton, driving his side forward from midfield as Norwich failed to establish a foothold in the tie.
Indeed, it was the Danish international who provided the cross for Kilbane to put Everton into the lead with a bullet header from three yards out.
James McFadden impressed on the right for the hosts, while teenage prodigy Wayne Rooney was a constant menace.
The young England striker wasted several gilt-edged opportunities to extend Everton's lead, the most glaring coming when he fired over from eight yards out with only the keeper to beat.
Rooney's profligacy gave Norwich renewed hope as the half progressed and they equalised when Brennan scored a wonderful left-foot volley from Paul McVeigh's free-kick.
Huckerby went close for Norwich
The Canaries seized the initiative and thought they had taken the lead through Roberts' cool finish, but the tall striker was adjudged to have strayed just offside.
Roberts' disappointment was compounded just before the break when he made contact with Everton defender Alan Stubbs in the Norwich area and referee Matt Messias pointed to the spot.
Ferguson converted the penalty and restored the hosts' advantage.
After the interval, Norwich searched for an equaliser and were denied twice by some smart goalkeeping by Nigel Martyn, who brilliantly blocked Darren Huckerby's goalbound effort.
Norwich stood firm in the face of Everton's own attacks, but the First Division side found themselves going further behind when Roberts again tangled with Stubbs in the area.
The outcome was the same as Ferguson converted his second spot-kick of the afternoon to send Everton into the fourth round of the FA Cup.
Everton: Martyn, Hibbert, Stubbs, Unsworth, Naysmith,
McFadden (Jeffers 64), Carsley, Gravesen (Yobo 45), Kilbane,
Rooney, Ferguson (Campbell 83).
Subs Not Used: Radzinski, Simonsen.
Booked: Naysmith, Hibbert, Yobo.
Norwich: Green, Edworthy, Fleming, Mackay, Drury,
Brennan (Henderson 72), Francis (Mulryne 80), Holt, McVeigh,
Roberts (Jarvis 80), Huckerby.
Subs Not Used: Crichton, Shackell.
Referee: M Messias (N Yorkshire).