Seagoe suffered due to early struggles in Saturday’s Marshall Cup derby defeat by 3-1.
The tie - switched to Lurgan Town due to a waterlogged Tarsan Lane - left Seagoe on the backfoot from the first whistle.
Sustained pressure resulted in a goal around 20 minutes.
Phil McCoy proved sharp between the posts and Seagoe’s resolute play produced a reward when captain Stuart Corkin linked up with Jasir Silva before Stephen Duke produced the finish.
Duke was left frustrated by a fine Lurgan Town save, Sam Nesbitt’s corner-kick created confusion and Andrew Gordon squandered a promising opportunity as Seagoe pushed forward.
Seagoe suffered the setback of a second goal and quickfire red card double.
Lurgan Town extended the lead to 3-1.
Best on the day for Seagoe were Chris Megaw, Alan Cantle and Geoff McCollum during the third-round exit.