Brees, who led the NFL in passing yards last season with 5,069, hit Jeremy Shockey with a 22-yard scoring toss in the first possession of the game. Brees completed 6-of-9 passes for 88 yards.
Brunell, in his 17th NFL season, launched a 64-yard touchdown pass to Robert Meachem midway through the third quarter to give New Orleans the lead for good. Garrett Hartley added a 54-yard field goal for the Saints early in the fourth quarter.
Cincinnati's only score came on a 14-yard throw from J.T. O'Sullivan to Chris Henry in the second period.
The Bengals obtained O'Sullivan as a free agent from San Francisco this year. He completed 9-of-11 passes for 100 yards after starter Carson Palmer had gone 7-of-11 for 133.
Notable deaths of 2014 [PHOTOS]
Arizona Cardinals player formally charged