Poor discipline repeatedly cost Chicago in the first half as the Bears allowed the Steelers to take an 11-point advantage into the break.
Pittsburgh built on their lead with two more field goals in the third, before a Chicago touchdown late in the quarter sparked a frantic final period.
Two more field goals gave the Steelers a seemingly insurmountable lead, until the Bears hit back with two quick touchdowns to go ahead 27-26 with 1:46 left on the clock.
However, veteran quarterback Ben Roethlisberger marched downfield to put Boswell in position to seal the game-winner.
Roethlisberger finished with 205 yards and two touchdowns, while Chicago counterpart Justin Fields had 291 yards to go along with a touchdown and an interception.
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