Junior outfielder Carson Kainer, who was named the Phillips 66 Big 12 Player of the Week on April 11, extended his season-best six-game hit streak to seven with an RBI double in the top of the first.
But the Bears responded by putting up one run in each of the 2nd, 3rd, 4th and 5th innings and kept the Horns from scoring until the top of the 5th when freshman Kyle Russell hit a solo homerun to right center.
In the top of the 6th, Russell kept Texas rolling with a two-RBI single, scoring Jordan Danks and Kainer, to tie the game at 4.
Junior OF Drew Stubbs broke the tie in the top of the 8th with a solo shot down the leftfield line to give Texas a 5-4 lead.
Nick Peoples added to the advantage with an RBI single in the 9th, scoring Russell for his second run.
But in the bottom of the 9th, Baylor 1B Tim Jackson hit a two-RBI single to tie the game at 6 and take it into extras.
In the top of the 13th, Harris hit the RBI single to put Texas up for good.
Baylor pitcher Ryan LaMotta pitched eight innings and finished with five earned runs and five strikeouts, but the loss went to Nick Cassavechia in four innings pitched.
Texas, conversely, used five pitchers, with freshman Austin Wood getting the win with a career-best 4-1/3 innings of shutout relief.
The Horns return from Waco to play the remaining two games of the series at Disch-Falk Field on Friday at 7:00 p.m. and Saturday at 2:00 p.m.