McHargue completed 12 of 21 passes for 144 yards and ran for another 68 yards. Charles Ross led all rushers with 117 yards on 16 carries as Rice (2-5) rolled up 301 yards on the ground.
While Rice's quarterback didn't throw for a touchdown, running back Luke Turner hit Donte Moore for a 38-yard TD pass in the second quarter. Two Chris Boswell field goals rounded out the scoring for Rice.
David Glasco II scored UTSA's sole touchdown on a 36-yard pass from Eric Soza. Glasco also ran it in for the 2-point conversion.
Kristian Stern had two field goals for the Roadrunners, who lost for the first time this season after a 5-0 start.