Having lost their opening six games, the Giants were efficient, if not overwhelming, in producing three takeaways to one lost fumble against the Vikings.
Peterson, the league's Most Valuable Player last season and a 2,000-yard rusher, was held to 28 yards five days after his two-year-old son was buried in South Dakota.
The man accused in the death was indicted on second-degree murder and manslaughter charges, prosecutors said on Monday.
Peterson was not helped by a rusty Josh Freeman, making his debut as Vikings quarterback after being signed as a free agent when Tampa Bay cut the 2009 first-round draft pick. Freeman frequently missed open receivers, and several of his throws sailed yards beyond his targets.
Josh Brown kicked three field goals and Rueben Randle caught a 24-yard TD pass for New York.
Marcus Sherels provided Minnesota's points with an 86-yard punt return, but his fumble without being hit during a runback midway in the third quarter gifted the Giants a opportunity they jumped at, with Peyton Hillis - signed last week - surging in from short range for a 17-7 lead.
The lead grew on Brown's 23-yard field goal to finish off a 16-play, 75-yard drive, and to 23-7 on his 36-yard kick.
Giants quarterback Eli Manning, who led the league with 15 interceptions, didn't throw one for the first time all season. And while Freeman was picked only one time, by Antrel Rolle, he looked like someone who hadn't played in more than a month.