League One Scunny beat the Rams on penalties after battling back from 3-0 and then 5-3 down to force extra time with two injury-time goals.
Northampton, who knocked out last season's runners-up Cardiff, have been drawn at home to Championship club Wolves.
Thirteen Premier League clubs were in the second round draw.
Five-time League Cup winners Aston Villa host Tranmere, while Paolo di Canio's Swindon, who beat Brighton on Tuesday night, travel to Stoke.
Swansea, QPR, Everton and Sunderland all received home draws, against Barnsley, Walsall, Leyton Orient and Morecambe respectively.
Fulham travel to Sheffield Wednesday, while Wigan head to the City Ground to face Nottingham Forest and West Brom are away at Yeovil.
Of the Premier League newcomers, Southampton travel to Stevenage, Reading host Peterborough and West Ham tackle Crewe at Upton Park.
Crystal Palace, last year's beaten semi-finalists, face Preston, while Crawley, who beat Millwall on penalties, host Bolton.
Gillingham host 2004 League Cup winners Middlesbrough, MK Dons play Blackburn at home, while Coventry welcome Birmingham to the Ricoh Arena. Oxford's victory over Bournemouth has earned them a trip to face Leeds at Elland Road.
Doncaster face East Yorkshire rivals Hull, Carlisle host Ipswich, Leicester meet Burton at the King Power Stadium and Plymouth play Burnley.
Draw in full:
Preston v Crystal Palace
Watford v Bradford
Swansea v Barnsley
Yeovil v West Brom
Coventry v Birmingham
West Ham v Crewe
Doncaster v Hull
Carlisle v Ipswich
Reading v Peterborough
Sheffield Wednesday v Fulham
Leicester v Burton
Burnley v Plymouth
QPR v Walsall
Stevenage v Southampton
Nottingham Forest v Wigan
Stoke v Swindon
Aston Villa v Tranmere
Crawley v Bolton
Gillingham v Middlesbrough
Northampton v Wolves
Everton v Leyton Orient
MK Dons v Blackburn
Leeds v Oxford
Sunderland v Morecambe
Norwich v Scunthorpe
The ties will be played during the week commencing August 27.
