Elliot Cadeau added 15 points and Seth Trimble had 12 points for the Tar Heels, who shot only 41.2% from the field. Jae'Lyn Withers had 10 points and 10 rebounds.
TJ Simpkins scored 18 points and Nick Dorn had 17 to lead Elon. Sam Sherry added 15 points and TK Simpkins had 14.
Elon used a 14-0 burst to take a 71-69 lead with less than 7 minutes to play. Davis hit a floater and then worked free for a corner 3-pointer as momentum shifted back to the Tar Heels.
The Tar Heels took a 32-21 lead on Withers' 3-pointer just under the eight-minute mark of the first half. North Carolina scored the final seven points of the first half for a 51-39 lead at the break, holding the Phoenix without a point in the last two minutes.
The Phoenix, coming off a 13-19 season, made eight of 10 shots during the second-half rally. Dorn made five 3-point baskets
The Tar Heels will have lots to work despite winning their 20th consecutive season opener. Davis, coming off an All-American season and beginning his fifth college season, has posted a double-figure point total in four season openers.
After falling behind, North Carolina clamped down defensively as the Phoenix went six consecutive possessions without scoring while the Tar Heels built an 80-71 edge.
North Carolina will need to tighten things up for its next game, a visit to No. 1 Kansas on Friday. Elon hosts Bluefield (Va.) on Saturday.
The Associated Press contributed.