Baker Mayfield drove Texas Tech 75 yards in the game's first 1:09 and hit Kenny Williams with a 50-yard scoring pass that put the Red Raiders in early control.
Mayfield, however, threw three interceptions in Texas Tech's hurry-up offense and TCU rallied into a 10-10 tie in the second half.
Webb was called on to save the game and he did so by marching the Raiders 75 yards in 10 plays after the Horned Frogs had tied the contest. Webb found Bradley Marquez with a 19-yard scoring pass in the opening Big 12 Conference game for both teams.
TCU punted the ball in the closing minutes while trailing by seven points and Texas Tech clinched the victory with 2:48 to go on a 37-yard field goal by Ryan Bustin.
Texas Tech remained unbeaten in three games while TCU fell to 1-2, its other loss having come against LSU.
Trevone Boykin completed 38-of-61 passes for the Horned Frogs, but was intercepted twice. TCU's tying touchdown in the fourth quarter came on a 17-yard run by B.J. Catalon.
Mayfield was 21-of-40 for 216 yards while Webb completed 3-of-4 for 77.