The contest was marred by a sideline collision during the first period in which New Orleans Coach Sean Payton suffered a broken leg. Assistant Joe Vitt took over Payton's duties and the Saints announced Payton will have surgery Monday.
Tampa Bay moved into a tie for the NFC South lead with New Orleans at 4-2.
Barth kicked a field goal in each quarter, the last of them coming from 40 yards for the game's final points with 7:24 to play.
Quarterback Drew Brees then led the Saints from their own 24 to the Tampa Bay 2-yard line, but on fourth down he was intercepted by Black in the end zone.
It was one of three interceptions for Brees in the game, spoiling a performance in which he threw for 383 yards and a score.
Tampa Bay quarterback Josh Freeman completed 23-of-41 passes for 303 yards and two touchdowns. He hit Arrelious Benn with a 65-yarder early in the second quarter and then found Preston Parker from 19 yards out with 3:40 to go in the opening half. Tampa Bay opened a 20-10 lead at intermission.
The collision that injured Payton involved one of his own players. New Orleans tight end Jimmy Graham caught a pass near the sideline before being tackled. Graham slammed into Payton as the tackle was being made.