Vikings wide receiver Stefon Diggs made two touchdown receptions in the first half.
Bradford completed 27 of 32 passes with no turnovers. Adam Thielen caught nine passes for 157 yards, and tight end Kyle Rudolph added a touchdown reception for Minnesota (1-0).
With Peterson on the opposite sideline in his New Orleans debut, Vikings rookie running back Dalvin Cook ran 22 times for 127 yards.
Peterson got the first two carries of the game and picked up a first down, but he was silent afterward as New Orleans -- the No. 1 offense in the league last season -- had little room to run and few open receivers.
Peterson finished with six carries for 18 yards in a timeshare with Mark Ingram and rookie Alvin Kamara.
Saints quarterback Drew Brees was 27 of 37 for 291 yards and one touchdown.
New Orleans (0-1) kicked field goals on its first four trips to the red zone until Brees connected with tight end Coby Fleener for an 8-yard touchdown with 1:56 left in the game.
Both teams traded long drives aided by penalties to open the game, but each settled for a field goal. The Saints added another field goal after a roughing-the-passer penalty on Tom Johnson after he sacked Brees.
Then Bradford got comfortable after getting good protection from his much-maligned offensive line and Minnesota took to the air against the New Orleans defense.
The Vikings had a three-play, 74-yard drive for the first touchdown. Bradford connected with Thielen for 35 yards and Jarius Wright for 21 before finding Diggs wide open in the end zone from 18 yards.
Minnesota appeared set to run out the clock when it assumed control with 1:43 left in the half, but after converting a first down, Bradford took advantage of holes in the secondary again. Thielen caught a pass out of the slot and ran for 44 yards.
Diggs did the rest. He caught a 20-yard pass as he was leveled by safety Kenny Vaccaro, who was penalized for unnecessary roughness. Three plays later, Diggs hauled in a pass one-handed along the sideline in the end zone for his second touchdown.
Peterson, the Vikings' career rushing leader with 11,747 yards, was quiet in his return. He ended the first half with just four carries for 14 yards.
NOTES: New Orleans RT Zach Strief left the game in the second quarter with a leg injury and didn't return. ... Longtime Vikings WR Randy Moss was honored during halftime in the team's Ring of Honor. Moss is second to former teammate Cris Carter in receptions (587), receiving yards (9,316) and receiving touchdowns (92) in his eight seasons in Minnesota. ... WR Stefon Diggs' two first-half touchdowns were the first for a Minnesota player in one half of the season opener since Moss accomplished the feat in 2004. ... With second-round RB Dalvin Cook and third-round C Pat Elflein starting, it marked the first time two rookies started on offense for the Vikings since 1963.