The Longhorns (2-0) also got 16 points from Andrew Jones, 12 from Kerwin Roach II, and 11 points apiece from Jacob Young and Eric Davis Jr. Texas outshot New Hampshire 47.5 percent to 31.7 percent, including 53.3 to 29.4 in the second half.
Tanner Leissner led New Hampshire (1-1), which competes in the American East Conference, with 14 points. Iba Camara added 12 points and 10 rebounds, and Jacoby Armstrong tallied 11 for the Wildcats before fouling out with 9:03 to play.
Texas led by as many as 14 points in the first half as Osetkowski racked up 12 points -- five of which came from the free-throw line -- and eight rebounds while playing all but one minute in the half.
Armstrong led the Wildcats with 11 points in the half.
New Hampshire cut the Longhorns' lead to 29-23 with two free throws by Armstrong with 3:45 to play in the first half. Texas responded by scoring 10 of the final 12 points in the period and took a 39-25 lead at intermission.
Any chance the Wildcats had to get back in the game was derailed in the first six minutes of the second half when Texas forged a 12-6 run and pushed their advantage to 51-31.
Texas played without freshman center Mohamed Bamba, who sustained a concussion during practice Sunday and continues to be evaluated as part of the program's concussion protocol.
On Tuesday, Bamba was named to the 50-player watch list for the John R. Wooden Award, which is presented at the end of the season to the college basketball national player of the year.