The Dolphins gave up two fourth-quarter touchdowns that rallied the Saints into a 21-16 lead, but Devlin drove Miami 92 yards in seven plays while using just 49 seconds.
Devlin, in his third NFL season, completed 22-of-38 passes for 259 yards with two touchdowns and two interceptions. He also hit Dion Sims with an 8-yard scoring pass in the first quarter.
New Orleans drove 80 and 68 yards to scores in the fourth quarter. The first touchdown came on a fumble recovery by Tim Lelito in the end zone after Khiry Robinson had lost the ball trying to score from the 1-yard line. Robinson gained 115 yards on 22 carries.
The Saints briefly took the lead when Ryan Griffin threw a 2-yard scoring pass to Austin Johnson with 11:04 remaining in the contest.
Miami, which played in the Hall of Fame Game to start the exhibition season, finished at 2-3. New Orleans lost for the first time in four games.
Miami gets going in the regular campaign with a trip to Cleveland on Sept. 8. New Orleans, with Sean Payton coaching his first regular-season game after serving a suspension for all of 2012, will be at home that day against Atlanta.