Army led 55-54 with 1:02 remaining after Turner made his first free throw but missed the second. Texas A&M teammate Elston Turner came up with the rebound, however, and Army's Dylan Cox fouled Harris on a 3-point attempt from the left side of the arc after the outlet.
Harris, with 43 seconds remaining, made all three of his free throws to lift the Aggies (9-3) to a 57-55 lead, one they wouldn't relinquish as Cox missed a lay-up and a 3-pointer on ensuing possessions.
Kyle Wilson paced Army (6-6) with a game-high 15 points.
The Aggies made only 15 of 27 free throws, but were solid from the 3-point line by making 8 of 20.