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.
NCAA title-game hero Kris Jenkins says he will withdraw his name for the NBA draft and will return to Villanova for his senior season.
Andy Katz breaks down the likelihood of Villanova, Syracuse, North Carolina or Oklahoma making a run to the Final Four next season.
Florida State has announced that Xavier Rathan-Mayes will return to the court for his junior season. The 6-foot-4 guard declared for the NBA draft last month but did not hire an agent.