Rhymes connected with a three-run dinger to left in the first inning Friday after back-to-back singles from rightfielder Mark Laird and shortstop Alex Bregman, giving the Tigers an early 3-0 lead.
Auburn answered in the second and third frames, but Katz increased the Tigers' lead to 5-2 with a two-run homer of his own to left, marking his 10th home run of the season.
Third baseman Christian Ibarra added two RBIs, giving LSU a 7-2 lead in the fifth inning.
A wild pitch and an RBI single from Katz gave the Tigers two more insurance runs in the eighth inning.
Bregman recorded his fifth three-hit game of the season and has now recorded a hit in 10 consecutive games. Rhymes finished 2-for-4 on the night and has reached base in all 22 games this season.
LSU starting pitcher Aaron Nola (3-0) got the win, striking out nine in 6.2 innings while giving up two runs on nine hits.
LSU's 20-2 record is the Tigers' best 22-game mark since the 1997 team also won 20 of its first 22 games.
The teams resume the series at 6:30 p.m. Saturday. Junior right-hander Ryan Eades will take the mound for LSU, while Auburn will counter with junior left-hander Michael O'Neal.