It was a tougher-than-expected challenge, with the No. 19 Tar Heels leading just 46-41 at halftime.
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Armando Bacot scored 25 points and grabbed 13 rebounds in the best season opener of his career.
RJ Davis and Cormac Ryan both scored 13 points, and Harrison Ingram had 12 points for the Tar Heels.
North Carolina won its 19th straight season opener.
Bacot, a preseason All-American, scored 13 first-half points. His 25 points in an opener beat the 16 points he had last year against UNC Wilmington.
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The Tar Heels had transfers Ingram (Stanford), Ryan (Notre Dame) and Paxson Wojcik (Brown) in the starting lineup. North Carolina is seeking a fresh start after last season began with a No. 1 national ranking and ended without a berth in the postseason except the Atlantic Coast Conference tournament.
DaQuan Smith and Kenyon Giles both scored 18 points for Radford, which was seeking its second victory against a ranked team in program history. Bryan Antoine added 13 points.
Cormac's 3-pointer from the corner in front of the North Carolina bench pushed the margin to 74-61, finally giving the Tar Heels some separation after Radford hung around and made some of the home fans uncomfortable.
North Carolina rattled off the last eight points of the first half for a 46-41 lead. Radford missed four straight shots and committed a turnover before the break.
North Carolina scored on its first seven possessions. Radford climbed back and it was 20-20 with less than 11 minutes to play in the half. The Highlanders later led 29-26.
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Smith scored 12 points as Radford held a 35-32 edge.
The Highlanders had spurts of offense that allowed them to keep it close deep into the second half. They didn't have the means to shut off the interior. Meanwhile, the Tar Heels demonstrated the ability to get the ball inside, with some strong passing from the perimeter setting up easy shots. Ryan is a needed perimeter threat, accounting for three of the team's seven 3-point baskets.
Up next for the Tar Heels is a visit from Lehigh on Sunday night.
The Associated Press contributed.