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.
Nike Men's West Virginia Mountaineers Blue Hyperelite ELITE Basketball Game JacketShop
Former Kentucky guard Ed Davender, the only Wildcat with at least 1,500 points and 400 assists, has died. He was 49.
Guard Naz Mitrou-Long, whose 2015-16 season ended in December due to hip problems, has been granted a hardship waiver for the coming season, Iowa State said.
As the Sooners look to life without superstar Buddy Hield and his sidekicks, it's clear their unseasoned players will need to grow up fast.