The Yorkshire side, who had already knocked Leeds United out of the competition and beaten the Championship sides Sheffield Wednesday, Wolves and Southampton in the Carling Cup, even saw their goalkeeper Andy Warrington save a late penalty from Matty Fryatt.
Darlington beat League Two rivals Bury at the Williamson Motors Stadium, with Alan White scoring the winner early in the second-half after the Shakers failed to clear a corner properly.
Tranmere Rovers also secured passage to the last eight of the competition with a 1-0 home win - with Ryan Shotton's header enough to see off Morecambe at Prenton Park.
Shrewsbury Town made the biggest mark in the southern section, hammering Dagenham & Redbridge 5-0 at New Meadow.
Luton Town beat Walsall 1-0, and Colchester United saw off Bournemouth by the same scoreline.
Brighton & Hove Albion were knocked out of the Carling Cup, losing 4-1 at home to the Championship side Derby County.