The Canaries started life in the third tier with a shocking 7-1 home defeat to local rivals Colchester last week - a result which saw Bryan Gunn become the first managerial casualty of the season when he was sacked on Friday.
Elsewhere, League Two new boys Burton host local rivals Chesterfield, while Gillingham face Colchester in another all-League One encounter.
Sixteen sides - Brighton, Charlton, Exeter, Hartlepool, Leeds, Leyton Orient, Southampton, Southend, Swindon, Tranmere, Aldershot, Grimsby, Macclesfield, Notts County, Port Vale and Shrewsbury - were handed first-round byes.
Johnstone's Paint Trophy, first-round draw:
Oldham v Accrington Stanley; Crewe Alexandra v Stockport County; Walsall v Bury; Morecambe v Carlisle United; Darlington v Lincoln; Rochdale v Bradford City; Rotherham United v Huddersfield Town; Burton Albion v Chesterfield; Barnet v Millwall; Norwich City v Brentford; MK Dons v Dagenham & Redbridge; Gillingham v Colchester; Wycombe v Northampton; Cheltenham Town v Torquay United; Hereford v Bristol Rovers; Bournemouth v Yeovil Town.
Matches to be played week commencing Monday August 31.