Romar Morris scored on two short touchdown runs and Jack Tabb caught a touchdown pass as the Tar Heels (7-6) closed the season by winning six of their final seven games under second-year coach Larry Fedora.
Cincinnati (9-4) was looking to become the bowl's first back-to-back champion since Virginia did it 10 years ago, but last year's MVP Brandon Kay was limited to 181 yards passing and no touchdowns.
The Tar Heels had five sacks, including one for a safety.
North Carolina came in having lost its previous three in-state Belk Bowl appearances, but bolted to a 23-3 halftime lead behind a pair of long touchdown drives led by Williams and Logan's nifty kickoff return.
The Tar Heels dominated from the opening kickoff, taking a 7-0 lead midway through the first quarter, effectively moving the ball 68 yards in 11 plays behind a no-huddle offense with Morris scoring on a 2-yard run.
Brandon Ellerbe and Kareem Martin sacked Kay in the end zone for a safety - the first of three sacks in the opening half.
On the ensuing kickoff, Logan put the Tar Heels in control by fielding the kickoff near the left sideline, getting a block on the inside and bolting up the middle of the field for a touchdown.
Williams made it 23-3 in the second quarter, hitting Tabb on a quick slant for a 3-yard touchdown strike.
Unlike last year's Belk Bowl when Cincinnati spotted Duke 17 points and stormed back to beat the Blue Devils 48-34 behind Kay's four touchdown passes, there would be no Bearcats comeback.
Switzer turned in another big special teams play in the third quarter when he fielded a punt at his own 14 and, after several Cincinnati overran the coverage, returned it 86 yards for a touchdown and a 29-3 lead. Switzer has five punt returns for a touchdown, tying him with Chad Owens of Hawaii (2004) for NCAA single-season record.
The Bearcats reached the end zone on a 15-yard touchdown run by Ralph David Abernathy, but the Tar Heels answered with a methodical 13-play, 65-yard drive, with Morris scoring his second touchdown on a 1-yard plunge to make it 36-10.
After Shaq Washington scored on a 10-yard run to cut the lead to 36-17, Cincinnati forced a fumble on the ensuing kickoff and recovered at the Tar Heels 6. But the Tar Heels mounted a goal-line stand and Kay's pass on fourth-and-3 fell incomplete, thwarting any comeback attempt.
James Hurst, North Carolina's first-team All-ACC left tackle, injured his knee in the first quarter and did not return.