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.