The second quarter passed without either side troubling the scorers, but the visitors were back on the scoreboard early in the third when Stafford looped a pass over to Calvin Johnson on the right side of the end zone for a four-yard score.
Robbie Gould reduced the deficit to four points midway through the third quarter with a 25-yard field goal, and another kick, this time from 32 yards, made it a one-point game in the fourth quarter.
The Lions surged eight points clear late in the fourth quarter as Stafford's 14-yard pass found Johnson deep in the end zone for his second touchdown of the game.
The Bears had one last chance to force overtime with under a minute to play when Marshall collected Josh McCown's pass for a touchdown.
The hosts went for the two-point conversion but Matt Forte could not make his way into the end zone with his run.