Trailing by three points, the Saints (9-3) took over at their own 32-yard line with 4:20 left. Brees led them on a seven-play drive, which included a 42-yard strike to Robert Meachem (106 yards on three grabs) and was capped by his 3-yard TD toss to Colston.
Brees ended with 313 yards on 24-of-29 passing and two touchdowns as New Orleans stretched its winning streak to five.
Chris Ivory added 117 rushing yards with two scores on 15 carries and Colston made three grabs for 65 yards in the victory.
Bengals signal-caller Carson Palmer gained 249 yards and a TD on 23-of-33 passing, with six going to Terrell Owens for 47 yards and a score.
Chad Ochocinco collected 96 yards on five receptions for the Bengals (2-10), who dropped their ninth straight.