The Clarets' hopes of an FA Cup run were clinically extinguished by a Norwich side riding the crest of a wave since their promotion to the top flight.
Despite the priority of continuing their impressive start to the season and maintain their Premier League status, the Canaries didn't take their foot off the gas as they handed Burnley their first third-round exit since 2008.
Jay Rodriguez's 13th-goal of the season gave Burnley hope as he pegged back Norwich's early 2-0 lead with a magnificent header.
But despite missing a penalty through Grant Holt, City added two further goals in the second half as Burnley's win-or-bust approach brought only the latter and left them to focus all their energies on their Championship challenge.
The Clarets had gone with about the most attack-minded side they could muster with Martin Paterson returning to the starting line-up alongside both Jay Rodriguez and Charlie Austin.
But the policy of fighting fire with fire threatened to unravel straight away as City struck twice in the opening 12 minutes.
Simeon Jackson heads Norwich 2-0 up
Burnley had actually started well with both Austin and Rodriguez forcing goalkeeper Declan Rudd to make nervy saves on a rare outing in the Canaries' goal as Paul Lambert made five changes from the Norwich side that had won last time out at QPR to climb to ninth place in the top tier.
The home side still boasted a potent attack, though, and they wasted no time in making it tell.
Collecting a half-cleared corner, Wes Hoolahan hung up a perfect cross towards the far post where there was always going to be one winner as Holt charged in to bury a header past Lee Grant.
And six minutes later Grant was again picking the ball out of the net as this time Simeon Jackson leapt to meet an Adam Drury cross and nod home.
Jay Rodriguez's header beats Declan Rudd
The Clarets were threatening to take on water but their bold policy paid off almost instantly as Rodriguez got his side back into the game with the third goal of a remarkable opening quarter of an hour.
The striker had plenty to do with a header from Ross Wallace's free-kick but from some 16 yards out he managed to get the power and direction to beat Rudd just inside his left-hand post.
With the tie back in the balance, the Clarets gave as good as they got as both sides continued to throw men forward with near gay abandon.
But they needed a slice of luck to reach the break just one goal behind thanks to Holt's penalty miss with half an hour played.
Austin was penalised for blocking an Andrew Crofts' shot with his arms but City's talismanic skipper fluffed the chance to get his second of the game as he steered his spot-kick tamely wide of the target.
Grant Holt misses from the spot
Austin almost made him pay as only a fine block from Daniel Ayala prevented the striker, back at the ground where he made his Football League debut with Swindon Town, sweeping home a cross from David Edgar.
But the major scares continued to come in front of the Clarets' goal and eight minutes into the second half the Canaries again almost doubled their lead.
Grant made an excellent save from Hoolahan but the ball fell for Holt, whose shot hit a post and rebounded to Jackson who stabbed wide from inside the six-yard box.
However, it proved only a temporary stay of execution as Norwich made it 3-1 with an hour gone.
City's movement had been top-class all afternoon and when Drury and Jackson combined to open up the Clarets' defence, missing the suspended Kieran Trippier at right-back, Surman ran on to clip a shot over the Grant.
Lee Grant tries to keep the Canaries out
Burnley, with Marvin Bartley on for the injured Wallace following a freak collision between Wallace and team-mate Ben Mee, tried to respond with Austin just failing to get a decisive touch to Edgar's teasing cross.
But Norwich looked a confident side and with 17 minutes to go they killed the tie off as substitutes Aaron Wilbraham and Steve Morison made an immediate impact.
Wilbraham's first touch forced Grant to save at full stretch, but with the ball running loose Morison couldn't miss as he tapped in his eighth of the season to guarantee City a place in the fourth round for the first time in five years.
Charlie Austin goes close to pulling one back
Burnley: Grant, Edgar, Easton, Mee, Amougou (MacDonald 76), Marney, McCann, Wallace (Bartley 52), Paterson, Rodriguez, Austin (Stanislas 71). Subs: Hines, Treacy, Jackson, Stewart.
Norwich: Rudd, Martin, Drury, Whitbread, Crofts, Holt (Wilbraham 71), Jackson (Morison 71), Surman, Hoolahan (Bennett 77), Fox, Ayala. Subs: Johnson, Lappin, Barnett, Steer.
Booked: Whitbread 80, Ayala 84
Referee: Michael Jones
Cup dreams over