Nicola Dragovic had 14 points and six rebounds for the Bruins (2-3), who had three players foul out. Malcolm Lee added 11 points and four assists.
UCLA, which trailed by nine points with 9:58 remaining, tied it at 67 on Jerime Anderson's 3-point shot with 8.6 seconds left.
On the ensuing inbound play, Hayward received Avery Jukes' pass, drove the lane under intense physical pressure, was fouled by Michael Roll and made both free throws.
Hayward then intercepted Anderson's long inbound pass intended for Dragovic in the Bruins' frontcourt as time expired.
Butler used a 12-3 surge to take a 21-11 lead with 9:14 left in the first half, and led by 14 points with 7:38 before halftime. Mack scored 16 points in the first half.