The minnows held on for the remaining 89 minutes thanks to dogged defending, poor finishing and a bit of luck.
Kidderminster Harriers were moments away from a famous win over Premiership Wolves, with a last minute equaliser from Alex Rae earning the visitors a replay at Molineux.
Second Division Blackpool were also denied a lucrative replay against Premiership side Portsmouth, Yakubu Ayehbeni's scrambled goal in stoppage time saw the South Coast side through to the Fourth Round.
All three non-league sides avoided defeat, with Accrington Stanley and Scarborough earning replays against Colchester and Southend respectively.
Yeovil Town were denied a creditable draw against Liverpool, with late goals from Emile Heskey and Danny Murphy seeing the Reds through.
Results in full:
Accrington Stanley 0 Colchester Utd 0
Aston Villa 1 Man Utd 2
Barnsley 0 Scunthorpe 0
Birmingham 4 Blackburn 0
Bradford 1 Luton 2
Cardiff 0 Sheff Utd 1
Coventry 2 Peterborough 1
Crewe 0 Telford 1
Everton 3 Norwich 2
Fulham 2 Cheltenham 1
Gillingham 3 Charlton 2
Ipswich 3 Derby 0
Kidderminster 1 Wolves 1
Leeds 1 Arsenal 4
Man City 2 Leicester 2
Mansfield 0 Burnley 2
Middlesbrough 2 Notts County 0
Millwall 2 Walsall 1
Northampton 1 Rotherham 1
Nottm Forest 1 West Brom 0
Portsmouth 2 Blackpool 1
Preston 3 Reding 3
Southampton 0 Newcastle 3
Southend 1 Scarborough 1
Sunderland 1 Hartlepool 0
Swansea 2 Macclesfield 1
Tottenham 3 Crystal Palace 0
Tranmere 1 Bolton 1
Tottenham 3 Crystal Palace 0
Wigan 1 West Ham 2
Wimbledon MK 1 Stoke 1
Watford 2 Chelsea 2
Yeovil 0 Liverpool 2
Numbers for Monday's Fourth Round draw:
1. Wimbledon or Stoke City
2. Sheffield United
3. Burnley
4. Telford United
5. Barnsley or Scunthorpe United
6. Portsmouth
7. Northampton Town or Rotherham United
8. Fulham
9. Manchester City or Leicester City
10. Newcastle United
11. Liverpool
12. Birmingham City
13. Nottingham Forest
14. Watford or Chelsea
15. Kidderminster Harriers or Wolverhampton Wanderers
16. Gillingham
17. Swansea City or Macclesfield Town
18. Arsenal
19. Tottenham Hotspur
20. Manchester United
21. Southend United or Scarborough
22. Sunderland
23. Tranmere Rovers or Bolton Wanderers
24. Preston North End or Reading
25. West Ham United
26. Middlesbrough
27. Ipswich Town
28. Coventry City
29. Everton
30. Accrington Stanley or Colchester United
31. Luton Town
32. Millwall
2. Sheffield United
3. Burnley
4. Telford United
5. Barnsley or Scunthorpe United
6. Portsmouth
7. Northampton Town or Rotherham United
8. Fulham
9. Manchester City or Leicester City
10. Newcastle United
11. Liverpool
12. Birmingham City
13. Nottingham Forest
14. Watford or Chelsea
15. Kidderminster Harriers or Wolverhampton Wanderers
16. Gillingham
17. Swansea City or Macclesfield Town
18. Arsenal
19. Tottenham Hotspur
20. Manchester United
21. Southend United or Scarborough
22. Sunderland
23. Tranmere Rovers or Bolton Wanderers
24. Preston North End or Reading
25. West Ham United
26. Middlesbrough
27. Ipswich Town
28. Coventry City
29. Everton
30. Accrington Stanley or Colchester United
31. Luton Town
32. Millwall