Greg Jennings helped the Vikings take the lead in the first quarter, collecting a short pass from Matt Cassel before breaking off tackles from Ike Taylor and Troy Polamalu to complete a 70-yard touchdown.
Bell added his second early in the third quarter as Pittsburgh tried to overtime the deficit, but Peterson and Jennings were also able to reach the end zone again to keep the Vikings ahead.
A Shaun Suisham field goal brought the Steelers to within seven points, allowing the game to push for a game-levelling touchdown late on.
Roethlisberger completed passes to Antonio Brown and Cotchery to leave them just short of the end zone, but the quarterback came under pressure before fumbling, handing the ball back to the Vikings with just eight second left.
The win of Minnesota's first of the NFL season, while Pittsburgh fall to 0-4.