Bolen's run produced the only touchdown of the contest, which left both teams with 2-2 records.
Josh Scobee kicked two field goals for Jacksonville, which begins its regular season at home against Denver. Trevor Harris was 8-of-12 for 86 yards.
Matt Bryant provided all the Atlanta scoring with three field goals. Atlanta will start its campaign in Pittsburgh.
Jacksonville won despite managing 100 fewer yards than the Falcons.