Senior guard Matt Mobley added 14 points and junior forward LaDarien Griffin had 10 for the Bonnies (26-7).
Adams, the co-Atlantic 10 Player of the Year, shot just 2 of 16 from the field while scoring eight points but came through in the clutch.
Junior guard Aaron Holiday had 20 points and seven assists before fouling out for UCLA (21-12). Bruins sophomore guard Prince Ali tallied 13 points and freshman guard Kris Wilkes added 10.
St. Bonaventure will face sixth-seeded Florida in Thursday's first round of the East Region in Dallas.
"It can't get better," Bonnies coach Mark Schmidt said in a postgame interview on truTV. "Our guys just fought. We had toughness, we made big shots. In 1970, (Hall of Famer) Bob Lanier got hurt and didn't have a chance to play UCLA (in the Final Four). This is for him."
Adams struggled through a miserable offensive game before swishing a 19-footer with 49.1 seconds remaining to give the Bonnies a 60-58 lead.
Holiday turned the ball over on UCLA's next two possessions, and Adams was fouled after each of them and made 3 of 4 free throws to make it a five-point margin.
Holiday was called for an offensive foul with 12 seconds left and Mobley added two free throws to seal it. Holiday wound up with 10 turnovers.
The Bonnies led by seven points early in the second half before UCLA rattled off seven straight and tied it on Ali's basket with 16:32 remaining.
The Bruins continued surging, and Holiday's 3-pointer gave UCLA a 49-44 lead with 9:16 left.
The Bruins again led by five a short time later but then went scoreless for 4:58 while the Bonnies ripped off 12 points in a row.
Stockard drained a 3-pointer, Griffin scored the tying basket, Stockard added a layup and Mobley knocked down a jumper to give St. Bonaventure a 55-51 lead with 5:35 left.
Mobley drained a straightaway 3-pointer to boost the lead to seven with 4:10 left.
Holiday hit a jumper to end UCLA's scoring drought with 3:24 left and Ali buried a 3-pointer with 2:31 remaining to move the Bruins within 58-56. Holiday hit two free throws to tie it with 1:11 left.
Stockard scored 13 first-half points as St. Bonaventure held a 28-23 lead at the break.
Stockard's production was timely with Adams missing his first eight shots while having two points on 1-of-9 shooting before halftime. Helping the Bonnies was that UCLA committed 11 turnovers in the opening half.
--Field Level Media
St. Bonaventure | UCLA | |
Courtney Stockard 26 | Scoring | Aaron Holiday 20 |
Jaylen Adams 3 | Assists | Aaron Holiday 7 |
Idris Taqqee 11 | Rebounds | Thomas Welsh 15 |
Courtney Stockard 6 | Free Throws Made | Aaron Holiday 2 |
Courtney Stockard 4 | Steals | Kris Wilkes 1 |
N/A | Blocks | Gyorgy Goloman 4 |
Team | Points | FG% | 3PM-3PA | FTM-FTA | Assists | Rebounds | Blocks | Steals | Turnovers |
St. Bonaventure | 65 | 38.3 | 4-19 | 15-21 | 7 | 30 | 0 | 11 | 6 |
UCLA | 58 | 42.0 | 10-30 | 6-10 | 13 | 37 | 6 | 1 | 20 |