The Rams (4-2) also blocked a punt for a score and a 10-point lead in the first half. Malcolm Brown returned the loose ball 8 yards for the franchise's first such touchdown since 2005.
The Jaguars (3-3) botched a decent chance to tie the game early in the fourth period when Blake Bortles fumbled on one play and then threw an interception on the next. It cost Jacksonville a shot at ending its up-and-down start to the season.
Los Angeles essentially sealed its third road victory on Greg Zuerlein's 29-yard field goal with 2:32 remaining. The Rams hadn't started 3-0 away from home since 2001.
The Jaguars, coming off an impressive victory at Pittsburgh, still haven't won consecutive games in more than a year and lost for the ninth time in 10 games at EverBank Field.
Rookie Leonard Fournette set the tone again by scoring on the first play from scrimmage. After Cooper's kickoff return, Fournette took a handoff and went 75 yards for his sixth rushing touchdown and seventh score of the season.
Fournette also scored on his 28th and final carry last week against the Steelers, a 90-yard run that put an exclamation point on a blowout. So Fournette totaled 165 yards and two scores on back-to-back carries.
His latest touchdown also made history. The consecutive scores marked the first time in the Super Bowl era that any game had two touchdowns of at least 75 yards in the first 30 seconds of play.
Fournette finished with 130 yards on 21 carries.
Todd Gurley ran 23 times for 116 yards for the Rams, who are in the middle of an 11-day road trip. Robert Woods caught five passes for 70 yards. And Jared Goff was 11 of 21 for 124 yards, including a 4-yard TD pass to Gerald Everett.
The Jaguars got two tries at a field goal in the third quarter during a strange sequence of penalties that ended up costing the Rams three points.
Jason Myers hooked a 41-yarder left, but Los Angeles was flagged for illegal formation. Jacksonville was penalized on the next play for a false start, moving the ball back to the original spot. Myers then drilled his second try from 41 yards.
Myers also missed twice from 54 yards.
Rams tight end Derek Carrier left in the third quarter and was evaluated for a concussion. Cornerback Trumaine Johnson left late in the third to be evaluated for a concussion, but returned a few plays later. Fellow cornerback Kayvon Webster briefly left the field a few plays later.
Rams: Will stay in Jacksonville for four days before flying to London and playing a "home game" against NFC West rival Arizona.
Jaguars: Play at AFC rival Indianapolis next week.
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