The win marked the Bruins' ninth straight victory, including five consecutive postseason games, and 16 of their past 17 overall.
Seven of the eight NCAA Super Regionals were in action on Saturday, and LSU and Stony Brook finally got a conclusion to an epic thriller that began on Friday night.
LSU's Mason Katz hit a walk-off RBI single in the bottom of the 12th inning on Saturday morning, after the game was suspended by rain earlier in the inning on Friday night.
That gave the Tigers a 5-4 victory in Game 1 of the best-of-three series, but Stony Brook responded with a 3-1 win in Game 2 to force a decisive Game 3 on Sunday.
Kevin Gausman, the fourth overall selection by the Orioles in Monday's First-Year Player Draft, pitched seven innings, allowing three runs. But the LSU righty was out-dueled by Stony Brook's Tyler Johnson, who allowed one unearned run in a complete game.
Gausman was the first college pitcher selected on Monday, going ahead of Stanford's Mark Appel, who was projected as a potential No. 1 overall pick. Appel's Cardinal were rained out on Saturday, and they will continue their series against Florida State on Sunday with the Seminoles leading, 1-0.
For Stony Brook, Travis Jankowski, taken by the Padres as the 44th overall selection on Monday, went 1-for-4. Going into the Draft, Jankowski was ranked No. 43 among available players by MLB.com's Jonathan Mayo.
Top-seeded Florida, along with South Carolina and Baylor, are all on the brink of advancing to Omaha after they each won the opening games of their respective series.
Led by catcher Mike Zunino, who was drafted third overall by Seattle, the Gators knocked off N.C. State in Game 1 of their series, 7-1.
Zunino homered and drove in three runs, while Hudson Randall pitched seven scoreless innings.
The Gamecocks' 5-0 victory over Oklahoma put them one win away from reaching what would be their third consecutive College World Series. Ace Michael Roth threw 7 2/3 scoreless innings and after a three-run second, the result was never really in doubt.
Logan Vick, who was drafted in the 11th round by the Indians, homered twice for Baylor, putting the Bears on the cusp of their first trip to Omaha in June since 2005. The Bears cruised past Arkansas, 8-1.
In the last game of the night, Kent State squeaked out an upset win over No. 5 Oregon in Game 1 of their series, 7-6.
Jumping out to a 5-2 lead after five innings, Kent State was led by shortstop Jimmy Rider and left fielder Alex Miklos, who each had two RBIs.
LSU 5, Stony Brook 4 (12 innings, Suspended in top of the 12th, completed Saturday)
Arizona 7, St. John's 6 (10 innings)
Stanford 1, Florida State 17
TCU 2, UCLA 6
Stony Brook 3, LSU 1
Arizona 7, St. John's 4
No. 1 Florida 7, North Carolina State 1
No. 4 Baylor 8, Arkansas 1
No. 8 South Carolina 5, Oklahoma 0
No. 2 UCLA 4, TCU 1
Kent State 7, No. 5 Oregon 6
Stanford at No. 3 Florida State (PPD)
Sunday's schedule (All times ET)
North Carolina State at No. 1 Florida, 1 p.m.
Stony Brook at No. 7 LSU, 1 p.m.
Arkansas at No. 4 Baylor, 4 p.m.
Stanford at No. 3 Florida State (if necessary), 7 p.m.
Oklahoma at No. 8 South Carolina, 7 p.m.
Kent State at No. 5 Oregon, 10 p.m.
Monday's schedule (All times ET)
North Carolina State at No. 1 Florida (if necessary), 1 p.m.
Arkansas at No. 4 Baylor (if necessary), 4 p.m.
Oklahoma at No. 8 South Carolina (if necessary), 7 p.m.
Kent State at No. 5 Oregon (if necessary), 7 p.m.