Cooper Kupp starred for the Rams, who have bounced back from losing last season's Super Bowl while the Bengals are still searching for their first win.
Josh Reynolds touched down to put the Rams back ahead in the second quarter before Joe Mixon gave the Bengals hope by levelling the score at 10-10.
But the Rams took control in the second half with a magnificent 65-yard touchdown from Kupp steering the favourites back into a seven-point lead at the half-time break.
Todd Gurley extended LA's lead on a three-yard touchdown run and in the process moved the Bengals closer to compiling the season's worst record.
In the day's other games, the Minnesota Vikings finished 19-9 over the Washington Redskins, the Seattle Seahawks beat the Atlanta Falcons 27-20 and the Los Angeles Chargers nailed a close one to end 17-16 against the Chicago Bears.
Mixed fortunes continued to beset the New York Giants as the Detroit Lions won 31-26, the Jacksonville Jaguars went 29-15 over the New York Jets and the Tennessee Titans won 27-23 at home in Nashville against the Tampa Bay Buccaneers.
The Philadelphia Eagles got back to some winning ways with a 31-13 takedown of the Buffalo Bills, the Indianapolis Colts did what was needed to eke out a 15-13 victory over the Denver Broncos and the New Orleans Saints demolished the Arizona Cardinals 31-9.
The largest number of offensive points in the round so far went to San Francisco, with the 49ers hammering the Carolina Panthers 51-13.
The Texans managed to beat the the Oakland Raiders 27-24 but it came at a cost, with defensive star JJ Watt potentially out for the rest of the season due a suspected torn pectoral muscle.
The score was closer than expected in Foxborough as the New England Patriots out-muscled the Cleveland Browns 27-13, while in Kansas City the hosting Chiefs lost 31-24 to the Green Bay Packers.