The Perth club completed the deal before the deadline for their Europa League second qualifying round first leg against Rosenborg in Norway next week.
Fallon was released by Aberdeen at the end of the season following a two-year stay at Pittodrie.
The 31-year-old hit just three goals in 37 Scottish Premier League appearances, although he struck six times in cup competitions for the Dons.
The New Zealand international has spent most of his career in England, spending time at Barnsley, Swindon, Swansea, Plymouth and Yeovil.
Fallon boosts Tommy Wright's striking options following the departures of Gregory Tade and Rowan Vine.