The defending NFL champion Eagles were on the ropes until the final play, when Joshua Karty had a 44-yard field goal attempt to win for the Rams, but Eagles defender Jordan Davis blocked the kick and returned it 61 yards for a game-clinching touchdown.
"It came down to the technique and fundamentals of everything," Davis said. "We knew his angle. We knew his launch point. We just hit the gap, put our hands up at the right time and then I seen the ball on the ground.
"I just took off from there. I wanted possession. I wanted to make sure we ended the game right. Didn't know I was going to go that fast, didn't know I was going to make it all the way to the end zone."
Hurts completed 21-of-32 passes for 226 yards and ran for 40 more, while Matthew Stafford threw for 196 yards and two touchdowns in a losing cause.
"It was just a matter of figuring it out," Hurts said. "We were gritty. We stayed together as a team. We did not quit. Nobody gave up. I'm very proud of that."
In another battle of unbeatens, Eddy Pineiro kicked a 35-yard field goal at the finish to give host San Francisco a 16-15 victory over Arizona.
Tampa Bay, Indianapolis, and the Los Angeles Chargers also improved to 3-0 with triumphs, while Green Bay and Cincinnati suffered their first defeats.
Philadelphia's Zack Baun intercepted a Stafford pass on the opening drive, and Hurts capped a 38-yard Eagles march on a one-yard "tush push" touchdown run.
Stafford completed a 44-yard touchdown pass to Davante Adams, and Karty kicked field goals of 28, 33, 46 and 51 yards to give Los Angeles a 19-7 half-time lead.
Moments into the third quarter, Hurts was sacked by Jared Verse, and Nate Landman recovered a fumble to set up a 10-yard Stafford touchdown pass to Kyren Williams.
But Hurts connected with Dallas Goedert on a 33-yard TD pass and hit A.J. Brown on a nine-yard scoring toss to pull the hosts within 26-21.
After Philadelphia's Jalen Carter blocked a 36-yard Karty field goal attempt, the Eagles drove 91 yards on 17 plays over 6:54 for Hurts to throw a four-yard touchdown pass to Devonta Smith with 1:48 remaining.
A Hurts two-point conversion pass failed, but the Eagles still led 27-26, setting the stage for the last-play field goal block.
"I'm not proud of how we started the game," Hurts said. "I'm not proud of the flow in which we played, the execution and focus we had early in the game, but I'm proud of everybody not giving up and being determined to find a way to win."
The New York Jets got a 50-yard touchdown return by Will McDonald off a blocked field goal attempt for a 27-26 lead at Tampa Bay, but Chase McLaughlin kicked his fifth field goal, a 36-yarder as time expired, to keep the Buccaneers undefeated with a 29-27 victory that left the Jets winless.
In Nashville, Jonathan Taylor ran for 102 yards and three touchdowns to power Indianapolis over Tennessee 41-20.
Cleveland's Andre Szmyt kicked a 55-yard field goal as time expired to give the Browns a 13-10 upset of visiting Green Bay, who squandered a 10-point lead in the last 3:38.
With Cincinnati quarterback Joe Burrow out for three months with an injury, the Bengals were thrashed 48-10 at Minnesota. Carson Wentz threw for two touchdowns, and Jordan Mason ran for two in the Vikings' romp.
Rodgers passes Favre on all-time list
Pittsburgh's 41-year-old quarterback, Aaron Rodgers, threw his 509th career touchdown pass, passing ex-Green Bay teammate Brett Favre for fourth on the all-time list, in a 21-14 win at New England. Rodgers hit D.K. Metcalf for the milestone score early in the second quarter.
Backup Marcus Mariota, in for hurt quarterback Jayden Daniels, threw for one touchdown and ran for another in host Washington's 41-24 victory over Las Vegas.
Travis Etienne's 10-yard touchdown run with 1:48 remaining gave host Jacksonville a 17-10 victory over winless Houston, while the Carolina Panthers took their first win, 30-0 over visiting Atlanta.
In later games Sunday, Arizona visits San Francisco in a matchup of 2-0 clubs with Dallas at Chicago, New Orleans at Seattle, Denver at the Los Angeles Chargers and Kansas City at the New York Giants in a meeting of winless teams.