Gurley, who had a 71-yard touchdown run, is the first rookie to open his career with four consecutive 100-yard games according to STATS with data available since 1991.
Tavon Austin caught a 66-yard touchdown pass and ran for a 2-yard score, compensating for an early lost fumble.
The Rams (4-3) are 3-0 against the NFC West for the first time since 2004.
The 49ers (2-6) were hit hard by injuries, losing Reggie Bush early, and haven’t scored a touchdown in two games.