Goalkeeper Danny Porteous saved three penalties as the Tangerine Army beat their Bradford opponents 3-1 on penalties after a 2-2 draw in normal time.
Scawthorpe have been drawn at home to Nottingham side Poet Young Boys in the third round.
Athletic boss James Kaye said he was proud of his players after Sunday’s win at Askern Welfare.
“I was happy with the victory but was disappointed not to get the game wrapped up in normal time,” he said.
“But I’m proud of the lads. They grafted in very difficult pitch conditions.”
Adam Watson missed from the spot early in the game after Shaun Mundy was brought down by the visiting goalkeeper.
Westwood Park broke the deadlock but Mundy, who scored twice in Scawthorpe’s 4-0 win over Newark-based side RHP Sports and Social in the first round, levelled before half time.
Will McGhie put the home side ahead following excellent work by Steffan Cairns but the visitors grabbed a late equaliser from a set piece.
Porteous then stepped up to the plate in the shoot-out, making three saves and leaving McGhie to score the winning penalty.
Scawthorpe will host Poet Young Boys on February 13.
Athletic qualified for England’s most prestigious Sunday League competition after winning the Doncaster Rovers Sunday League Premier Division title last season.
They are South Yorkshire's sole representatives in this year’s 64-team tournament.