After Garrett Hartley tied the opening exhibition game for both teams on a 23-yard field goal with 2:40 to play, the Patriots drove 65 yards in eight plays under the guidance of rookie quarterback Zac Robinson.
Robinson threw a short pass to Darnell Jenkins on a third-down play from the New England 36 and Jenkins turned it into a 52-yard gain to set up the winning field goal.
New England never trailed, but the Patriots let a 24-7 lead get away when Hartley tied it with his fourth-quarter field goal.
Laurence Maroney scored on a 5-yard run and a 1-yard-run for the Patriots and Larry Beavers started the New Orleans comeback with a 97-yard kickoff return after Maroney's second score.
Super Bowl-winning quarterback Drew Brees was 9-of-13 for 55 yards.
New England's Tom Brady was 5-of-8 for 67.