Johnson snapped a 6-6 tie with a two-yard touchdown toss to tight end Alge Crumpler with 3:11 left in the third quarter.
Michael Vick, Atlanta's projected starter at quarterback, was seven of 12 for 58 yards and carried twice for 11 yards.
Duckett, the Falcons' first-round draft pick, busted a 25-yard run. The former Michigan State star has had an abbreviated training camp since signing a week ago.
Mark Brunell went three of eight for 28 yards for the Jaguars, who pulled within 16-13 on backup quarterback Gary Garrard's 14-yard run with 5:25 to go in the final period.
Garrard completed four of eight passes for 41 yards and had 47 yards on five rushes.
2014: The Year in Music [PHOTOS]