Mike Davis gained 115 yards on three receptions for Texas (1-0).
Taylor McHargue managed just 79 passing yards on 14 completions for Rice (0-1).
The Longhorns managed to grab a 13-3 edge in the second quarter with a one-yard touchdown run by Cody Johnson and a 23-yard Tucker field goal.
Boswell nailed a 49-yard field goal in the final minute of the first half, cutting the Owls' deficit to 13-6 at intermission.
With just over nine minutes left in the third quarter, Boswell drilled his third field goal of the contest, this time a 26-yarder that cut the Texas advantage to 13-9.
The Longhorns put together an impressive drive on the ensuing possession, marching 72 yards in eight plays. John Harris threw a 36-yard touchdown strike to Jaxon Shipley to push Mack Brown's team ahead by a 20-9 margin.
Fozz Whittaker put the game away for the Longhorns in the fourth quarter by reaching the end zone twice. After a seven-yard touchdown run, Whittaker hauled in a 26-yard scoring strike from Gilbert.
Texas finished with 506 total yards, more than double the amount amassed by Rice (224).