Bradford hit Lance Kendricks with a 44-yard touchdown pass in the first period, and a 1-yard scoring run by Keith Toston in the third quarter put the Rams in front for good.
Lewis completed 12-of-15 passes in a backup role and ran three times for 15 yards. Kendricks caught three passes for 73 yards.
First-round Jacksonville draft pick Blaine Gabbert was 9-of-18 for 132.
Jacksonville (1-3) begins the regular season at home against Tennessee. St. Louis, which joined Detroit as the only unbeaten teams in the exhibition campaign, hosts Philadelphia.