Meeks was 9 of 13 from the field as taller and more athletic Tar Heels outshot Davidson 47 percent to 33 percent.
North Carolina held a double-digit lead for the entire second half, then began to pull away late.
The Tar Heels outrebounded the Wildcats 51-42.
Peyton Aldridge had six 3-pointers and finished with 25 points to lead Davidson (2-1). Tyler Kalinowski added 12 points for the Wildcats.
The game was played at Time Warner Cable Arena, home of the NBA's Charlotte Hornets, who are owned by former Tar Heel standout Michael Jordan.
Davidson, which is making the transition from the Southern Conference to the Atlantic 10 this season, stayed close with North Carolina early on behind some solid outside shooting by Aldridge, who was 6 of 10 from 3-point range.
But North Carolina began to build a comfortable double-digit lead late in the he first half with an 11-0 run.
North Carolina finally pulled away with about five minutes left, stretching its lead to 79-60 behind Meeks and Jackson, who dominated inside. North Carolina outscored Davidson 44-22 in the paint.
The Tar Heels finally put the game away the 6-foot-9 Meeks scored on a run out and was fouled for a 3-point play.
The Tar Heels put an exclamation point on their 37th win in 41 meetings over Davidson when J.P. Tokoto slammed home a high-flying one-handed dunk on a pretty feed from Jackson.
North Carolina: Brice Johnson, who came in averaging 17.5 points per game, was a factor in the first half for the Tar Heels with nine points. But he didn't score in the second half and fouled out with about five minutes left in the game.
Davidson: The Wildcats came in having made 25 of 50 3-point shots on the season against Campbell and Catholic. They were 11 of 36 from beyond the arc on Saturday.