Bradford went 19-of-34 through the air, hooking up with Austin Pettis on a pair of scoring strikes as the Rams (2-3) snapped a three-game losing streak.
Pettis finished with four passes for 49 yards while rookie Zac Stacy rushed for 78 yards on 14 carries in his first NFL starting assignment.
The hapless Jaguars (0-5), meanwhile, dropped their 10th straight overall dating back to last season.
Blaine Gabbert completed 9-of-19 for 181 yards, a touchdown and two interceptions before leaving with a left hamstring injury early in the third quarter.
Justin Blackmon, returning from a four-game substance-abuse suspension, led all receivers with 136 yards and a touchdown on five catches.