Conference National side Luton were in control from the outset at Kenilworth Road after Matthew Barnes-Homer scored in the fifth minute.
Corby Town, who play in the Conference South, were put further on the back foot when striker Nathan Jarman was handed a straight red card in the 25th minute, and Luton soon capitalised on their advantage with Kenyan Taiwo Atieno scoring four minutes later.
Atieno bagged his second in the 62nd minute before Claude Gnakpa added a fourth with nine minutes remaining, before Glenn Walker and Chris Hope scored late consolations for Corby.
In Wednesday’s other first-round replay, Conference side York City upset League Two’s Rotherham United 3-0 at Bootham Crescent.
After a lacklustre first half, Chris Smith’s opener for York on 66 minutes sparked his side into action, with Michael Rankine netting a brace to secure a passage into the second round.
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