The Anteaters (38-23) advance to the super regional round for the fourth time in school history and will face Oklahoma State.
Five of the eight national seeds were ousted after a run of regional upsets.
The Beavers joined No. 2 Florida, No. 4 Indiana, No. 5 Florida State and No. 8 LSU as regional losers. The number of national seeds that lost in regionals matches 2007 for the most since the NCAA went to its current tournament format in 1999.
UC Irvine took the lead with two runs on three hits in the sixth. Palmer singled home Chris Rabago and Adam Alcantara was hit by a pitch with the bases loaded. The Anteaters added a run in the seventh.
Evan Manarino (4-3) allowed one run on two hits in 3 1-3 innings of relief for UC Irvine.
Jake Thompson (3-2) gave up three runs on four hits in the loss for the Beavers (45-14).