Harris, the league's leading scorer, was 4 of 5 from 3-point range and the Mountaineers (9-5) improved to 3-0 against the Horned Frogs since both schools joined the Big 12 last season.
Amric Fields had a season-high 22 to the lead TCU (9-4) in his first Big 12 game. He missed most of the last season with a major knee injury that kept him out of the first three games this season. He recently returned from a four-game absence because of a broken left hand.
Brandon Parrish scored 18 points and had four 3-pointers, including one to get TCU within two with 3:06 remaining.
Kyan Anderson, TCU's leading scorer, finished with 12 points before fouling out with 6:26 remaining.
Staten hit back-to-back buckets to keep West Virginia in front in the final 3 minutes and made a breakaway layup to clinch the win in the waning seconds after TCU missed a pair of 3-pointers.
West Virginia built a lead early in the second half and stayed in front by dominating the offensive boards. The Mountaineers had three second-chance baskets in the first few minutes after halftime, with even the little guys getting involved.
Gary Browne, a 6-1 guard, missed a layup but tipped the ball in from the other side of the rim. With an eight-point lead down to two, the 6-3 Harris ended up with the ball for an easy layup and a 54-50 lead.
Terry Henderson had another put-back basket for a 65-60 lead as the Mountaineers had 17 offensive rebounds to eight for the Horned Frogs.
TCU was in the double bonus for the last 8 minutes but missed several key free throws, including two by Hudson Price with the Horned Frogs down 65-62 with 4:26 remaining. The Frogs finished 10 of 17 at the line.
Neither team led by more than five points in a high-scoring first half, with Harris and Fields getting 13 apiece.
TCU had the biggest lead at 28-23 after consecutive 3-pointers from Parrish.
West Virginia went ahead 36-35 when Juwan Staten found Harris for a long alley-oop dunk, and Nathan Adrian's 3-pointer with 1:12 left in the first half gave the Mountaineers a 42-39 halftime lead.
Harris was 3 for 3 from 3-point range in the first half, although West Virginia shot just 39 percent as a team.
The Frogs were 7 of 10 from long range before halftime, with Fields, Parrish and Anderson each going 2 for 3.