After a jumper by Carter gave the Mountaineers (10-13, 6-7) a 65-62 lead with 36 seconds remaining, Chris Long was unable to hit a 3-pointer that could have lifted the Phoenix (10-14, 4-9) into a tie. Sims then made two free throws for a five-point lead, 67-62, with 10 seconds left.
Sims, the Mountaineers' career points leader, went over the 2,000 mark and now has 2,015.
Appalachian State trailed 36-30 at halftime but outshot Elon in the second half 50 percent (12 of 24) to 30.4 percent (7 of 23). The Phoenix led by as many as eight points in the second half before Mitch Woods' 3-pointer -- his only points of the game -- put Appalachian State ahead to stay, 61-59.
Lucas Troutman scored 19 points and Long 15 for Elon.
Citing "an abundance of caution," NC State has pushed its trip to Italy to 2017 after the U.S. State Department warned of "the risk of potential terrorist attacks throughout Europe."
Four-star power forward Royce Hamm ended his recruitment on Friday with a commitment to head coach Shaka Smart and the Texas Longhorns after also considering Vanderbilt, Arkansas, Baylor and Houston among others.
Wisconsin's Nigel Hayes did his best Stone Cold Steve Austin impression and brought the pain to some fellow Badgers.