Demarcus Holland and Javan Felix each scored 12 points for Texas (7-1). Felix went 2 of 3 from beyond the arc and Kendal Yancy contributed 11 points and six rebounds.
The Longhorns led by as many as 13 points in the second half and were able to hold off Vanderbilt (4-3), which trimmed the Longhorns lead to one point in the final minute after an Eric McClellan steal and dunk.
Felix sunk a pair of key free throws to make it 65-62 before Holland's three-point play with 12.6 seconds left sealed the win.
Austin native McClellan scored a team-high 22 points for Vanderbilt while John Henderson scored 12 points on 5 of 8 shooting and Rod Odom had 11 points.
Texas needed second-half comebacks in four of its previous six wins this year but started the game on an 8-2 run and never looked back. The Longhorns led for the final 39 minutes of the game, using a 16-3 run to take a 31-25 halftime lead.
Vanderbilt trimmed Texas' lead to three points when Henderson knocked down a jumper to make it 57-54 with 2:30 left to play.
Then, Taylor helped the Longhorns take control of the game down the stretch. He completed a three-point play and hit a floater following a steal by Holland all in less than 30 seconds to give Texas a 62-54 lead with 1:47 to go.
The Longhorns initially struggled from the free throw line, making just 10 of 16 attempts before knocking down five straight in the final 24 seconds.
Texas, whose victory Monday came in the first-ever Big 12-SEC challenge, has now won 46 consecutive home games against unranked non-conference opponents. The Longhorns have also won seven of their first eight games for the first time in three years.