The hosts, who began the season 5-0, saw their diminishing playoff hopes slip further away towards the closing stages of the first quarter when Peterson made it four consecutive games with a touchdown by powering over from a yard out.
Tevin Coleman, in for the concussed Devonta Freeman, gifted the ball back to the Vikings with a fumble shortly afterwards, but they just about survived when Teddy Bridgewater's pass into the endzone was intercepted by Ricardo Allen.
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The Falcons got their first points on the board from the resulting possession, although they were forced to settle for just three as Shayne Graham - signed in place of the injured Matt Bryant just three days ago - knocked over a 31-yard kick.
Minnesota did have the chance to restore their seven-point lead before half time when Captain Munnerlyn intercepted Matt Ryan's deep pass down the middle, but Blair Walsh's 51-yard field goal in the dying seconds of the half went wide left.
Atlanta finally found their way into the endzone in the third quarter, only for Coleman's one-yard rush to be nullified on a clipping penalty, while Ryan's pass towards Jacob Tamme in the endzone shortly afterwards was intercepted by Terence Newman.
That missed opportunity was punished by the Vikings, but once again only three points were added to the score as Walsh converted a 29-yard kick before nudging his side 10 points ahead with a 39-yard effort in the fourth.
The victory was all but wrapped up with less than five minutes of game time remaining as Peterson doubled his personal tally for the day and made it eight for the season with a 35-yard rush into the endzone.
Peterson's performance was enough to lift him up to 18th in the NFL's all-time rushing yards list, but the Falcons were not giving up in their bid to avoid a fourth straight defeat and pulled a score back through Nick Williams late in the fourth.
It was too little too late, however, as the Vikings moved to 8-3 for the season, while the Falcons' poor form leaves them on 6-5.