The score came after the Red Raiders (3-0, 1-0 Big 12) thought they'd scored on a 49-yard TD pass from Webb to Deandre Washington, but he dropped the ball at the half-yard line. An unsportsmanlike conduct penalty took them back to the TCU 14. Two plays later, Webb hit Marquez.
TCU (1-2, 0-1) scored its only touchdown when B. J. Catalon ran in from 17 yards to make it 10-all with 6:28 remaining.
TCU's Trevone Boykin went 23 for 36 for 194 yards and two interceptions.
Baker Mayfield, Texas Tech's starting quarterback, left the game in the fourth quarter with what appeared to be a leg injury. He went 21 of 40 for one touchdown and 216 yards. He threw three interceptions.
Webb went 3 for 4 for 77 yards and a touchdown.
"That's about as ugly as you can win, but we'll take it," Texas Tech coach Kliff Kingsbury said.
Ryan Bustin made a 37-yard field goal to put the game out of reach with 2:43 remaining.
The Frogs had their chances. A 69-punt return for an apparent touchdown by Brandon Carter in the fourth quarter was called back after officials ruled he'd signaled for a fair catch.
Carter had another punt return wiped out -- a 28-yarder in the third quarter on a clipping penalty.
Mayfield threw his third interception early in the fourth quarter off a tipped ball and into the hands of Derrick Kindred. But TCU got no points off it when Jaden Oberkrom missed wide right on a 55-yard field goal try.
TCU put together its first sustained drive of the game early in the second half but got only a field goal to pull within 10-3 with 6:01 remaining in the third quarter. The Frogs looked in good position for another score after Carter returned a punt to the Red Raiders 46. But a clipping penalty by TCU's Dominic Merka moved the ball back to the TCU 41 and the Frogs failed to get a first down and punted.
Boykin scrambled for a first down on a 30-yard run to the Red Raiders 32 late in the third quarter, but Jackson Richards intercepted his pass shortly after and returned it 20 yards.
Jace Amaro caught nine passes for 97 yards to lead Texas Tech. Eric Ward, the Red Raiders' leading receiver coming into the game, went without a catch. He was covered most of the night by Jason Verrett, TCU's standout defensive back.
Texas Tech went up 10-0 in the first quarter, but both teams' second quarters were plagued with penalties and punts by both teams.
Mayfield found Kenny Williams in the flat along the near sideline on the Red Raiders' initial series, and he picked up some blocks and evaded at least one tackler to score a 50-yard touchdown. After the Red Raiders stalled at TCU's 22 on their third possession, Bustin kicked a 39-yard field goal to put Texas Tech up 10-0.
Mayfield threw interceptions in each quarter, but TCU failed to capitalize on either. Boykin was stymied on the ground and through the air and failed to get the Horned Frogs going.
Mayfield and the Red Raiders had chances to build on their lead. They had first downs at the TCU 29, 49 and 36, but came away without any points.
The third of those opportunities in TCU territory came after defensive back J.J. Gaines intercepted Boykin's pass and returned it 23 yards to the TCU 36. Three penalties pushed them back to the Frogs' 48 and the Red Raiders punted.
Defensive end Devonte Fields, in his first start of the season after sitting out all but one quarter of a two-game suspension for an unspecified violation of team or university policy during the offseason, appeared ineffective for TCU.