Harris drained seven of 10 shots while Dawson added 13 points, six rebounds and five steals for the Spartans (2-1), who won their second straight.
Michigan State, which dropped its season opener against Connecticut, held Texas Southern to just 21.1 percent shooting.
Fred Sturdivant led the Tigers (1-3) with 10 points and 12 rebounds.
Chipotle plans first price increase in 3 years
Justin Bieber crashes Drake Bell's album release party