Rian Lindell's field goal with 34 seconds left had given the Buccaneers a 17-15 lead, but Smith (24-of-38, 256 yards, one touchdown, one interception) found Kellen Winslow with a 25-yard completion with time running down.
He then picked up 10 yards on a scramble with the Bucs' Lavonte David flagged for hitting out of bounds, setting up Folk's kick with 2 seconds left in the season opener.
The rookie signal-caller also led the Jets (1-0) with 47 rushing yards on six carries.
Josh Freeman completed 15-of-31 passes for 210 yards, one TD and one pick-off for Tampa Bay (0-1), which committed 13 penalties.
Celebrity Couples of 2014 [PHOTOS]