Bradford hooked up with Danny Amendola on a 2-yard TD play shortly before halftime to put the Rams up 10-3 and made it 17-3 with 10:52 left in the game on a 23-yard scoring strike to Daniel Fells.
Josh Brown booted a pair of field goals and Brandon Gibson caught six passes for 67 yards for St. Louis (4-4), which benefited from four Carolina turnovers and yielded only 201 yards of total offense.
Matt Moore was picked off three times for Carolina (1-6). He finished the game 23-of-37 passing for 194 yards and a scoring pass to Brandon LaFell.
Steve Smith grabbed nine passes for 85 yards in the Panthers' setback.