LSU coach Ed Orgeron made the announcement after Thursday's practice, adding that fellow freshman quarterback Max Johnson will play as well.
"It could have been a flip of the coin," Orgeron said of the battle between Finley and Johnson.
Finley, who is 6-foot-6, is a former four-star prospect from Ponchatoula, Louisiana.
Brennan, a redshirt junior, suffered the lower-body injury against Missouri on Oct. 10. He'd thrown for 1,112 yards, 11 touchdowns and three interceptions through three games.
Orgeron said Brennan threw the ball a little bit during Wednesday's practice and that he will dress on Saturday. But Orgeron added that Brennan is not expected to play.
LSU, which went undefeated and won the national championship last season, has gotten off to a sluggish start, losing two of its first three games.
Last weekend's previously scheduled game against Florida was postponed after the Gators had an outbreak of COVID-19 on the team.