Brandon Taylor scored 14 points and Delon Wright added 12 and the fifth-seeded Utes beat No. 4
The Utes (26-8) will travel to Houston to play the winner of Sunday's game in Charlotte, North Carolina, between top-seeded Duke (30-4) and No. 8 San Diego State (27-8).
Utah hasn't advanced to the Sweet 16 since 2005, when the Utes fell as the No. 6 seed to No. 2 Kentucky. The Utes are making their first tournament appearance since 2009.
L.J. Peak led Georgetown (22-11) with 18 points.
Jordan Loveridge's long jumper gave Utah a 57-53 lead late, but the Utes couldn't quite pull away from the physical Hoyas. After Peak's layup brought Georgetown within 61-57 with 3:06 left, Taylor nailed a three-pointer for Utah.
Dakarai Tucker's layup stretched Utah's lead to 66-59 at the 1:40 mark and Georgetown couldn't catch up.
The Utes opened the tournament on Thursday with a 57-50 victory over No. 12 seed Stephen F. Austin, while Georgetown advanced with an 84-74 victory over No. 13 Eastern Washington.
The Hoyas haven't been to the Sweet 16 since 2007, when they won as the second seed until dropping to top-seeded Ohio State in the Final Four.
Georgetown was making its 30th appearance in the tournament, including a national championship in 1984. The Hoyas settled for an
invitation last year after getting bounced from the 2013
as a No. 2 seed by No. 15 Florida Gulf Coast.
During the 1990s and into the 2000s, the Utes were a perennial NCAA tournament participant under coach Rick Majerus, advancing in 11 of his 14 seasons at the helm. In 1989, Utah got past defending national champion Arizona and landed in the title game but fell to Kentucky, 78-69.
The Utes won a national championship in 1944.
Utah saw the return of 7-foot center Dallin Bachynski, who missed the tournament opener after tweaking his ankle earlier in the week. Utah needed him against the bigger and heavier Hoyas.
The game featured contrasting styles: Georgetown's Princeton offense against Utah's motion offense. At first the Utes appeared to have trouble adjusting to the Hoyas, who found early success on the perimeter.
Tied at 32 at the half, Utah hit consecutive three-pointers to go up, 40-36, early in the second half. Taylor's three-pointer put the Utes up, 43-36. The Utah bench went wild, and Coach Larry Krystkowiak signaled the players to mute the celebration.
Isaac Copeland made a 3 for the Hoyas to cut into the lead, and Utah was hurt by a scoring drought that lasted more than 5:30 before Loveridge made a 3-pointer to give the Utes a 46-42 lead with 11:30 to go.