Moore, who has an NCAA-best 18 saves, surpassed the previous mark of 17 held by Blair Erickson and Eric Pettis.
UCI rallied with four runs in the ninth inning to win the first game, 4-2, then broke a 1-1 tie in the eighth to claim a 4-1 triumph in Game 2.
Grant Palmer (two for three), Chris Rabago (two for five) and Connor Spencer (one for four) drove in runs in the second game, in which Mitch Merten earned the win in relief of starter Elliot Surrey (five hits and one run in 6 1/3 innings).
Moore allowed one hit in the ninth to seal the deal.
In the opener, Jonathan Munoz, Jerry McClanahan, Grant Palmer and Evan Cassolato had RBIs to cap a four-run ninth. Connor Spencer was two for three.
Starting pitcher Andrew Morales allowed two earned runs on five hits in seven innings and left in line to absorb the first loss in his two seasons as an Anteater.
But Matt Fielding and Moore each pitched a shutout inning to earn the win and the save, respectively, allowing Morales to remain 17-0 at UCI.
Palmer drove in the game-winner in each game.
UC Davis is now 17-22, 3-11.
The series concludes on Sunday.
Host Vanguard scored in the 13th inning to earn a split with Concordia on Saturday and finish in a three-way tie for third place in the Golden State Athletic Conference with Concordia and Westmont.
The Lions (31-19, 21-15 in conference) wind up in a three-way tie for third with Concordia and Westmont, but by virtue of a tiebreaker process, get the the No. 3 seed in the four-team GSAC Tournament.
Westmont is the No. 4 seed and Concordia does not get a tournament bid.
After a fielder's choice forced the would-be go-ahead run at the plate for the first out in the Vanguard 13th, John Schwer singled to left field to plate the winning run.