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.
Paul Quinn College's new basketball court has beauty and an important history lesson
Paul Quinn, a historically Black college, unveiled an impressive new court design with historical significance.
Time to hit the reset button for Aggies basketball
Texas A&M's Buzz Williams says that after a year disrupted by the COVID-19 pandemic, it is time for his team to start over.
Wheeler, Howard share lofty goals at Kentucky
Newcomer Sahvir Wheeler, veteran Rhyne Howard are both looking for big seasons from their respective UK basketball programs.