County, led by former Blades defender Keith Curle, saw off Scunthorpe to book their place in Saturday’s draw while United received a welcome bye in the first round.
The tie is due to be played during the week commencing October 8.
Elsewhere in the Northern Section, Doncaster Rovers begin their campaign with a home tie against Chesterfield. Both teams were relegated last season from the Championship and League One respectively.
And in the Southern Section, troubled Portsmouth are at home to Wycombe Wanderers while Southend United and Dagenham meet at Roots Hall in an Essex derby.
The Shrimpers saw off AFC Wimbledon while John Still's Daggers claimed an impressive victory over Stevenage to book their places in Saturday morning's draw, which came on the same day the two sides meet at Roots Hall in League Two action.
Southend also claimed a 3-1 victory over Dagenham at the same stage of the competition last season.
Draw in full:
Shrewsbury Town v Crewe Alexandra Rochdale v Bury Morecambe v Preston North End Walsall v Port Vale Doncaster Rovers v Chesterfield York City v Coventry City Notts County v Sheffield United Hartlepool United v Bradford City
Portsmouth v Wycombe Wanderers Cheltenham Town v Oxford United Torquay United v Yeovil Town Plymouth Argyle v Aldershot Town Northampton Town v Colchester United Leyton Orient v Barnet Brentford v Crawley Town Southend United v Dagenham and Redbridge