Dan Evans also won his first-round match, but Harriet Dart and Heather Watson went out.
Fitzpatrick carded a closing three-over-par 73 to miss out on the play-off between eventual winner Justin Thomas and Will Zalatoris by two shots.
Financial pressures mean the relegated club are certain to lose players but keeping captain Ben Mee could be crucial
Offiah won four Challenge Cup titles with Wigan between 1992-95.