Thomas put in 16 points and Barton had 14 for the Tigers, who turned back a late Samford rally with an 11-3 run to finish the contest.
Joe Jackson chipped in with 13 points and four assists for the Tigers.
Memphis now travels to Nassau, Bahamas, for the next round, where it will face Virginia Commonwealth Thursday.
The Bulldogs (1-3) were led by Tim Williams with 21 points.
Celebrity Families of 2014 [PHOTOS]