Goals from Shaun Wright-Phillips and Didier Drogba killed the game off and two late strikes from Michael Essien and Andriy Shevchenko buried the Championship side.
But they did have a couple of efforts and hit the woodwork twice themselves.
Chelsea boss Jose Mourinho, with one eye on Wednesday's Champions League clash away to Porto rang the changes for this FA Cup tie.
He dropped £32million Andriy Shevchenko to the bench and restored wingman Wright-Phillips to the starting line-up for the first time since facing Wycombe Wanderers on January 10.
Other changes saw Ghana midfielder Essien and Frenchman Claude Makelele also reduced to substitutes.
It was Wright-Phillips who had the first effort on goal, on four minutes, but his right-footed drive from 20 yards was weak and drifted harmlessly to goalkeeper David Marshall.
Frank Lampard produced a trademark rasping free-kick on eight minutes, which forced Marshall into a fine save.
Marshall's game ended two minutes later when he appeared to twist an ankle on the soft and uneven turf and was replaced by Paul Gallagher.
Norwich should have taken the lead on 18 minutes when a speculative shot by Dickson Etuhu beat goalkeeper Petr Cech on his right but hit the post. The ball ran back into play and Lee Croft had the simplest of chances to slot home from ten yards
But fortunately for the Blues, Khalid Boulahrouz scooped the ball away from the open goal.
John Terry was forced into a last-ditch headed save on 26 minutes to foil Darren Huckerby after an uncharacteristic mix-up between Terry and Boulahrouz almost let in the striker.
Lampard produced a fine shot from the edge of the Norwich area on 30 minutes, but substitute goalkeeper Gallagher was equal to it, by diving to his left and clearing the ball.
But Chelsea finally broke through on 39 minutes. Following sustained pressure a shot by Lampard was blocked and the ball fell to Wright-Phillips, on the edge of the penalty area.
The England winger took one touch before letting fly with a smart right-footer. The ball took a slight deflection off Andy Hughes and sent Gallagher the wrong way as the ball zipped into the top left-hand corner of his net.
Chelsea took just five minutes of the second half to double their lead. Drogba was right on the six-yard line when an Arjen Robben corner saw the ball fall invitingly to him.
He took a touch and swivelled around towards the goal and tucked the ball home with a toe-poke from his right boot past Gallagher and two other Norwich defenders on the goal-line.
Lampard was unlucky not to notch the third on 60 minutes but his free-kick, just outside the Norwich penalty area, went over the crossbar.
That was the last real action for the England midfielder as Shevchenko replaced him nine minutes later
Norwich had a sniff of goal on 75 minutes when a deep right-wing cross was met on the head of Chris Brown, but the ball zipped past Cech's right upright.
Cech was forced into two excellent saves with ten minutes remaining. The first was a superb tip away of a Croft blaster from 12 yards, which went clear after hitting the crossbar.
The second, he got low and saved the ball after substitute Martin rifled in a low left-footed drive from just inside the penalty area.
With five minutes remaining Cech was again in action, this time diving low to stop a glancing header by Dion Dublin from six yards out.
Right on time, Essien popped up from his centre-back position and tapped the ball home with his left foot from three yards out after excellent wing play by Shevchenko.
The final goal was the last action of the game when Shevchenko had the easiest of chances to nod home from the six yard line from Salomon Kalou's left-wing cross.