The Lions' offer sheet included a two-year deal, according to the Rams, so Brown will remain under contract through the 2020 season.
The Rams had placed an original-round tender on Brown, whom they acquired as an undrafted free agent in 2015.
Brown has served as Todd Gurley II's trusted backup the past two seasons and as a standout on special teams.
Last season, Brown rushed for 212 yards in 43 carries and caught five passes for 52 yards and a touchdown before he suffered a clavicle injury in Week 13 against the Lions that sent him to injured reserve.
Brown's injury played a significant role in the Rams' decision to sign C.J. Anderson, who added a late-season spark for the Super Bowl-bound team.
Brown could see increased playing time behind Gurley next season as the Rams continue to work through a process in determining how much of a workload Gurley can shoulder. Last season, Gurley was sidelined for the final two games of the regular season because of inflammation and soreness in his left knee, which was surgically repaired in 2014.
Rams coach Sean McVay said Gurley's knee certainly went through some "wear and tear" and "pounding" as he rushed for 1,251 yards and scored a league-best 21 touchdowns in 14 regular-season games.
The Rams also have expressed interest in re-signing Anderson in free agency.