The Browns went into the game missing eight starters and their entire first-team secondary for the prime-time matchup against the defending NFC champions.
Slot receiver Cooper Kupp caught 11 passes from quarterback Jared Goff, gaining 102 yards on those receptions and scored two touchdowns for the Rams.
The Browns forced two Goff interceptions but the team's offense still fell short for much of the game.
Rams star defensive tackle Aaron Donald hit the field after being listed as questionable during the week with a back issue.
Los Angeles started without right guard Austin Blythe and tight end Tyler Higbee.
The Associated Press contributed to this report.