Swilling Jr., son of former Georgia Tech and NFL linebacker Pat Swilling, made two 3-pointers in 19 minutes off the bench for the Golden Hurricane (5-3).
Drew Wilson had 11 points and 11 rebounds for the Bears (2-6).
Missouri State never led after 15:14 remained in the first half but was within eight points with eight minutes to play. But 3-pointers by Scottie Haralson and Swilling were the dagger, igniting an 11-2 run that gave Tulsa control.
The Bears shot just 29.4 percent from the floor, including 6 of 24 in the second half.
Shaquille Harrison had 10 points and seven rebounds for the Hurricane, whose first-year coach, Danny Manning, scored 42 points for Kansas against Missouri State in the second round of the 1987 NCAA tournament.