Moore delivered a walk-off single in the bottom of the ninth to capture the team's 11th series win.
Grant Watson made his UCLA record-setting 59th start, but the team fell to Gonzaga.
David Berg made his 162nd appearance to set a new NCAA record for career appearances.
The Bruins are 13-4 all-time against the Bulldogs and 15-5 in non-conference play this season.
Kevin Kramer recorded his 20th multi-hit game and Cody Poteet struck out a season-high seven.