Mejias-Brean was selected by the Reds in the eighth round of Major League Baseball's First-Year Player Draft, while Gonzalez was chosen by the Dodgers in the 27th round.
Mejias-Brean's big hit was enough for Wade, who threw the 79th complete-game shutout in College World Series history. The righty allowed five hits, struck out four and walked zero.
"It's an understatement for me to say he was outstanding," Arizona head coach Andy Lopez said. "He was pretty doggone good."
"He really, really pitched well tonight," UCLA head coach John Savage said. "He pitched as well as anybody we've seen all year."
UCLA (48-15) falls to the loser's bracket and plays Florida State (49-16) at 8 p.m. ET on Tuesday.
Tuesday's winner faces Arizona (45-17) at 4 p.m. on Thursday.
Stony Brook -- the fan favorite and second-ever No. 4 seed to advance to Omaha -- ends its season at 52-15.
"It's a hard loss, but I look back on the season, and I think we did what no one thought we could ever do, what everyone thought was basically impossible," said Stony Brook third baseman William Carmona, the Phillies' 11th-round pick. "And we made it happen somehow. And I'll never forget that."
The nightcap turned quickly, as the Wildcats went from zero hits to four runs in the fourth inning. UCLA right-hander Nick Vander Tuig was rolling -- with four strikeouts in 3 1/3 innings -- before three straight singles loaded the bases.
Mejias-Brean hit a two-run single to center -- advancing to second on a throwing error by UCLA center fielder Beau Amaral (seventh round, Reds) -- before Bobby Brown followed with a two-run double.
"I thought I made some good pitches. They put some good swings on it," said Vander Tuig, who only allowed one hit outside of the fourth. "I messed up a couple times, which hurt me."
Wade, who didn't allow a hit until UCLA's Pat Valaika's two-out single in the fifth, only faced one stressful inning. Cody Regis and Kevin Kramer followed Valaika's single with ones of their own before Kevin Williams flied out to left to end the inning.
Wade's performance Sunday was far different from when he faced Arizona in April and gave up six runs (five earned) on 10 hits in three innings.
"I got hit pretty hard the last time I pitched against them, and I knew they took advantage of the mistakes I made that game," Wade said. "I was really eager for the opportunity to throw against them today."
The opening game also quickly turned, as the top of Florida State's lineup -- stacked with draftees -- all but ended Stony Brook's Cinderella season in the third.
Center fielder James Ramsey followed consecutive two-out hits by third baseman Sherman Johnson before second baseman Devon Travis knocked out his own RBI single.
Stony Brook could have ended the inning on a grounder to the shortstop, but Cole Peragine short-armed his throw to first, allowing two more runs to score. After a walk, Gonzalez launched right-hander Brandon McNitt's 1-0 offering into the left-field bullpen.
"You can't, against good teams, make the mistakes we made and expect to survive that," Stony Brook head coach Matt Senk said. "And that's pretty much the bottom line."
Travis (13th round, Tigers) went 3-for-3 with two walks and a two-run home run, while Johnson (14th round, Angels) went 2-for-5 with two runs scored. Ramsey (23rd overall, Cardinals) finished 2-for-5.
"If I'm not mistaken, we got them all with two outs, and that's something that you just credit the young men for having great at-bats and getting it done," Florida State head coach Mike Martin said. "It's really encouraging to see our young men have this type of success against a very good baseball team."
Florida State 12, Stony Brook 2
Arizona 4, UCLA 0
Kent State (46-19) vs. No. 1 Florida (47-19), 5 p.m. ET
Arkansas (45-20) vs. No. 8 South Carolina (46-17), 9 p.m. ET.