Cunningham carried the ball 13 times as the Rams (5-6) piled up 258 rushing yards en route to the victory.
Zac Stacy added 87 yards on 12 carries with a score and Tavon Austin also ran for a TD.
Josh McCown completed 36-of-47 passes for 352 yards with two touchdowns and an interception for the Bears (6-5), who held onto a share of the NFC North lead despite the loss.
Brandon Marshall led Chicago's receivers with 10 grabs for 117 yards and a score.
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