It was the fifth interception of the night for Cutler, one of which came after the Bears had reached the San Francisco 1-yard line and another of which set up the only touchdown of the game.
San Francisco (4-5) halted a four-game losing streak and Chicago (4-5) lost for the fourth time in five contests.
Despite Cutler's problems, the Bears still had a chance to pull out of a victory when they took over at their own 20 with 2:47 to play. Chicago kept the final drive alive when San Francisco was called for a fourth-down, illegal contact penalty and Cutler eventually marched the Bears to the 49ers' 22 with 17 seconds left.
Cutler hit Devin Aromashodu with a 10-yard pass but then threw an incompletion in the end zone. With 8 seconds showing on the clock, he tried to hit Greg Olsen for the winning score, but Lewis grabbed it instead.
The game's lone touchdown was produced on a 14-yard run by Frank Gore with 2:03 left in the first half and came one play after Tarell Brown returned an interception 51 yards.
Robbie Gould kicked field goals of 50 yards just before the half and 38 yards in the third quarter to bring the Bears within one. Joe Nedney had a 21-yarder for the 49ers prior to the final Chicago drive.
San Francisco's Alex Smith completed 16-of-23 passes for 118 yards. Gore rushed for 104 yards on 25 carries.
Cutler was 29-of-52 for 307 yards. He has thrown 14 touchdown passes this season and been intercepted 17 times.