Jordan Bone added 16 and Jordan Bowden 14 for Tennessee (30-5), which advanced to play Iowa — winner of a first-round game over Cincinnati — on Sunday.
The Raiders (24-11) erased Tennessee's 42-30 halftime lead with an 11-2 run, tied the score midway through the second half, and kept the game close after that.
Schofield hit back-to-back 3-pointers to give Tennessee a cushion with 45 seconds left, and Bone and Schofield followed with a pair of foul shots each to extend the lead as Colgate ran out of time.
Jordan Burns had 32 points for Colgate, the most allowed by a single Tennessee opponent this season. The Raiders' 15 3-pointers matched the most allowed by the Vols.
This was a good start and a relief for Tennessee, which lost in the first round last year. The Vols were a No. 3 seed last year and 11th-seeded Loyola of Chicago got a basket off the glass with 3.6 seconds left and a 63-62 win.
Tennessee: With an advantage in size and speed, the Vols looked as if they might cruise but let Colgate climb back in. Three-pointers by another opponent hurt Tennessee. And it was just the third time this season that Tennessee's Grant Williams was held to 10 points or fewer.
Colgate: The Patriot League champions lost in their first tournament appearance since 1996 and third time overall. Colgate, with about 3,000 students in Hamilton, New York, still is seeking its first tournament win.
Tennessee moves on the second round to play Iowa on Sunday.