Cappelen, the first player from Denmark to win on the tour, had an 18-under 262 total after opening with rounds of 66, 65 and 65 at Crestview Country Club. The 24-year-old former University of Arkansas player earned $108,000 and full tour exemption for the rest of the season.
"It's what I always dreamed about," Cappelen said. "Coming over here and trying to be the first Danish person to make it over here in America, standing out in any way I can. Trying to be the first at something is what I want to do."
Cappelen two-putted for par from just off the 18th green to become the 21st Monday qualifier to win a Web.com Tour event and the first since 2011.
"It's amazing," Cappelen said. "It's kind of a dream come true for me right now. It's what I needed to do in this moment."
Weibring shot a 64.