Grant Holt and Simeon Jackson put Norwich 2-0 up in the opening 12 minutes before Jay Rodriguez hit back for Burnley almost immediately.
Holt then struck the post with 31st-minute penalty but the miss did not matter in the end as Norwich added two more goals in the second half through Andrew Surman and Steve Morison.
Hull edged into the fourth round at the expense of fellow npower Championship side Ipswich after winning 3-1 at the KC Stadium, while Sheffield United beat Salisbury 3-1 to book their place in the next round.
Tottenham claimed a comfortable 3-0 win over Cheltenham at White Hart Lane with goals from Jermain Defoe, Roman Pavlyuchenko and Giovani Dos Santos.
Craig Forsyth scored twice as Watford beat Bradford 4-2 at Vicarage Road.
Troy Deeney headed Watford ahead early on but James Hanson's close-range strike brought Bradford level five minutes later. Marvin Sordell edged the Hornets back in front before Forsyth's double sealed the win ahead of Nahki Wells' late score.
Simon Cox scored a hat-trick as West Brom also claimed a 4-2 win, beating Cardiff.
Derby beat Crystal Palace 1-0 thanks to Theo Robinson's ninth-minute goal while Marvin Emnes scored his first goal in nine games to help Middlesbrough defeat League Two Shrewsbury 1-0.
Elsewhere, there were goalless draws in the games between Dagenham and Redbridge-Millwall, Nottingham Forest-Leicester and Birmingham-Wolves.
Source: PA
Source: PA