Rotherham United defeated non-league Bromley on Saturday to advance to the next round while Harrogate Town also beat non-league opposition as they came from behind to beat Wrexham.
Doncaster Rovers beat Scunthorpe United in the early kick-off while FC Halifax Town defeated Maidenhead United in an 11-goal thriller at the Shay Stadium.
Championship and Premier League clubs enter the competition in round three. For sides reaching the third round, another £34,000 in prize money will be awarded.
Bradford City are still in with a chance of reaching the second round after they were held to a 1-1 draw against Exeter City at Valley Parade.
Sheffield Wednesday take on Plymouth Argyle on Sunday afternoon while York City and Guiseley were both knocked out on Saturday.