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.
Maryland point guard Melo Trimble, considered a lock to leave the program before struggling in the second half last season, has withdrawn from the NBA draft and will return for his junior season.
Louisville's Chinanu Onuaku, who recently underwent a medical procedure to correct a heart rhythm issue detected during the combine, will keep his name in the NBA draft pool.
Andrew White III will return to the Nebraska men's basketball team next season after removing his name from the NBA draft.