LSU, which won two of three games in the SEC series, improved to 25-7 overall and 8-4 in conference play. Florida fell to 25-7 overall, 7-5 in the league.
The SEC home series loss for Florida was its first since March 25-27, 2011, when the Gators dropped two of three to South Carolina.
Bonvillain (2-0) entered the game in the fifth inning with LSU trailing 7-5 and was brilliant, limiting Florida to no runs on one hit in 4 innings with one walk and 5 strikeouts.
Freshman first baseman Tyler Moore led LSU's 15-hit attack with a career-best 4 hits, and he scored one run. Second baseman JaCoby Jones contributed a 2-run double, and reserve outfielder Arby Fields contributed a run-scoring triple to highlight LSU's effort at the plate.
"This win demonstrates the character and determination we have on this ball club," LSU coach Paul Mainieri said. "We battled through all types of adversity with injuries and a five-run deficit, but we stayed together and everyone contributed to a great victory. To win a series against a talented team like Florida in their own ball park is a tremendous accomplishment, and I'm extremely proud of the young men wearing LSU across their chests. They were never intimidated and never deterred from the goal of winning this game (Saturday)."
LSU played the game without outfielder Raph Rhymes, the SEC's leading hitter, who suffered a mild concussion when he was hit in the head by a pitch in Friday's game. The Tigers also lost the services during Saturday's game of starting left fielder Alex Edward, who suffered an leg injury and was removed in the second inning.
The Tigers also won the series with a short-handed pitching staff as regular SEC starter Aaron Nola was held out of action with muscle fatigue in his throwing shoulder.
Florida closer Austin Maddox (2-2) was charged with the loss Saturday as he allowed one run on five hits in 2.1 innings with no walks and two strikeouts.
LSU took a 1-0 lead in the first when Jones was hit by a pitch, stole second and advanced to third on a throwing error by catcher Mike Zunino, and scored on designated hitter Grant Dozar's groundout.
The Gators responded with two runs in the bottom of the first when right fielder Preston Tucker lined an RBI double, and the Gators grabbed the lead when LSU center fielder Jared Foster misplayed DH Brian Johnson's fly ball, allowing Tucker to score.
LSU tied the game at 2-2 in the second when Tyler Hanover was hit by pitch, moved to second on Moore's single and scored on a throwing error by Florida reliever Steven Rodriguez.
The Gators regained the lead in the bottom of the inning on an RBI grounder by center fielder Daniel Pigott.
Florida erupted for four runs in the third against Eades as shortstop Nolan Fontana highlighted the inning with a three-run homer, his sixth of the year.
LSU responded with three runs in the fourth to narrow the gap to 7-5. Jones lined a 2-run double and right fielder Mason Katz doubled home Jones to cut the Florida lead to two runs.
The Tigers tied the game at 7-7 in the seventh when Fields tripled home a run and scored from third on a wild pitch by Maddox.
LSU grabbed the lead in the eighth as Dozar, Katz and shortstop Austin Nola contributed consecutive singles against Maddox to load the bases, and catcher Ty Ross lifted a sacrifice fly to score Dozar with the go-ahead run.
Nick Goody earned his fourth save of the season, retiring the Gators in order in the bottom of the ninth.
The Tigers return to action at 6:30 p.m. Tuesday when LSU plays host to Alcorn State in Alex Box Stadium.