Morgan made the winning catch on a pass from Chase Daniel, who went 11-of-18 through the air for 88 yards.
Drew Brees started behind center for the Saints and 17-for-25 for 179 yards and two touchdowns.
Lance Moore made a scoring grab as New Orleans improved to 2-2 in the pre-season.
The Texans -- who fell to 2-1 -- were led by Matt Schaub, who completed 15-of-18 passes for 194 yards and a touchdown.
Notable deaths of 2014 [PHOTOS]