The Ospreys shot 50 percent (30 of 60) from the field and 42.9 percent (9 of 21) from beyond the arc. Smith led the offensive charge, draining six 3-pointers in 11 attempts.
BaeBae Daniels had five of the Ospreys' nine blocks.
North Florida carried a 40-36 advantage into halftime and opened the second half with a 9-2 scoring run as the Ospreys outscored the Tigers 39-29 in the final 20 minutes.
North Florida had lost each of its past six season openers.
Edward Waters (1-2) committed 21 turnovers and made just 11 of 32 shots in the second half. The Tigers were outscored off the bench 48-10, including a 32-2 discrepancy in the second half.
ESPN's Dan Dakich joins The Paul Finebaum Show to discuss the SEC's presence in the NBA Draft.
Ted Kapita was the top-rated recruit from West Virginia in the class of 2015 and recently played at DME Academy in Florida.
With the Suns drafting Tyler Ulis, they now have five guards on their roster who attended Kentucky.