The Saints outscored the Colts 25-7 in the second half.
Porter's touchdown, a 74-yard sprint after picking off a Peyton Manning pass, came with 3:12 remaining in the game, killing a potential game-tying Colts drive.
Garrett Hartley kicked field goals of 46, 44 and 47 yards for New Orleans, which took the championship in its first Super Bowl appearance.
"We just believed in ourselves," Brees, who was named the game's MVP, said. "We knew that we had an entire city, maybe an entire country behind us. I'm trying to imagine what this moment would be like for a long time. It's even better than expected."
New Orleans and Indianapolis, the top seeds in their conferences, both finished the season with 16-3 records.
The Colts were seeking their second Super Bowl win in four years. They beat the Chicago Bears 29-17 in the 2007 title game.
Indianapolis put the Saints in an early 10-0 hole, led 10-6 at the half and 17-16 heading into the fourth quarter.
Brees completed 32-of-39 passes for 288 yards with a 16-yard touchdown throw to Pierre Thomas and a 2-yarder to Jeremy Shockey that sent the Saints ahead to stay with 5:42 to play.
Manning accumulated 333 yards passing on 21 completions in 45 attempts. The Colts' scores came on a 19-yard Manning pass to Pierre Garcon, a 4-yard run by Joseph Addai and Matt Stover's 38-yard field goal.
2014: The Year in Music [PHOTOS]