Brees went 29-of-35 through the air while Thomas finished with 91 yards of total offense as the Saints (5-0) won their 13th straight game under Coach Sean Payton, who missed the 2012 season while serving a year-long suspension.
Jimmy Graham made 10 receptions for 135 yards in the win.
Jay Cutler completed 24-of-33 passes for 358 yards with two TDs for the Bears (3-2), who dropped a second straight.
Alshon Jeffery led Chicago's receiving corps with 10 grabs for 218 yards and a score.
Swim Week Miami Beach 2014 [PHOTOS]