Beasant played for the full ninety minutes as the Cottagers went down 2-0 at Aldmondvale, Argentinian striker Fernando Pasquinelli scoring twice for the Scottish Premier League outfit.
The 44-year-old keeper was offered a trial by Fulham after rejecting a new contract at Brighton and travelled to Austria for the Cottagers' pre-season tour.
Beasant did enough to impress new Fulham manager Chris Coleman and the club are now keen to sign him as back-up goalkeeper to first-choice Edwin van der Sar.