Aaron Ross and Niem Stevenson each chipped in 15 points for the Red Raiders (15-6). Anthony Livingston totaled 12 points on four 3-pointers.
"We knew coming in it would be a hard fought contest for us here," LSU head coach Johnny Jones said. "(Texas Tech) coach (Chris) Beard has done a tremendous job this year with his basketball team, his program, and has gotten off to a really good start."
Smith led Texas Tech in rebounds with eight and Justin Gray followed with seven.
Antonio Blakeney led LSU (9-11) with 23 points. Duop Reath scored 12 with 11 rebounds for the double-double, his fourth this season. Wayde Sims contributed 10 points off the bench.
The Red Raiders let it fly from beyond the arc converting on 12 of 23 attempts.
"I kept thinking they were going to go on another run and I didn't feel safe in that game till about a minute left," Beard said. "It goes back to Johnny; he's not going to let his team quit."
LSU struggled from the 3-point line shooting 9 of 21 by the final buzzer.
The Tigers outmatched Texas Tech in points in the paint for a majority of the game until the Red Raiders took over, 22-20.
Texas Tech committed 10 turnovers to LSU's 12.
"It's hard for me to visualize beating good teams like LSU or Big 12 teams where we have one or two guys show up offensively," Beard said. "We have to have three or four guys show up every night. Tonight we got it done with four people in double-figures."