TCU defeated the Longhorns for only the second time in their last 30 meetings in what was their first battle since the Horned Frogs joined the Big 12 Conference.
Tucker scored on a 2-yard run in the first period and on a 1-yard plunge in the second, staking TCU to a 14-3 halftime lead.
A pair of second half field goals by Jaden Oberkrom lifted TCU's advantage to 20-6 and the Horned Frogs survived a late Texas offensive flurry created by backup quarterback Case McCoy.
McCoy drove the Longhorns 81 yards in nine plays and Jeremy Hills finished off the march with an 8-yard run to bring Texas within seven with 3:07 to play.
Texas quickly forced a TCU punt and the Longhorns took over at their own 34 with 1:44 to go, but on the second play of the drive McCoy was intercepted by Sam Carter.
It was one of four turnovers by Texas, three of them suffered by starting quarterback David Ash in the first half.
The Horned Frogs advanced to 7-4 for the season and are 4-4 in the Big 12 with a game remaining against Oklahoma. Texas (8-3, 5-3) had a four-game winning streak snapped.
TCU quarterback Trevone Boykin threw only nine passes but completed seven of them for 82 yards. He added 77 rushing yards as part of the Horned Frogs' 219-yard ground attack.