The Rams (4-3) snapped a three-game losing streak in the debut of cornerback Jalen Ramsey.
The Falcons (1-6) suffered their fifth straight loss and had quarterback Matt Ryan leave the game with an ankle injury. Ryan's right leg bent awkwardly as he was sacked by Aaron Donald and fumbled early in the fourth quarter.
Ryan limped off the field and was escorted to the medical tent on the sideline. He walked to the locker room, still favoring the ankle, with 7:36 remaining in the game.
There was no immediate update on the severity of the injury.
Donald was credited with a sack, forced fumble and recovery on the play. It was the Rams' fifth sack of Ryan, matching the five sacks by Atlanta's defense this season - none in the last four games.
Matt Schaub replaced Ryan and threw a 10-yard touchdown pass to Austin Hooper with less than three minutes remaining.
The final insult for Atlanta came with only 11 seconds remaining when Kenjon Barner fumbled a punt return when hit by Troy Reeder and Darious Williams recovered in the end zone for a touchdown.
Ryan's injury was the latest and perhaps most damaging dose of bad news for the woeful Falcons. Coach Dan Quinn's job security appears shaky at best as he quickly runs out of time to avoid a second straight losing season.
Atlanta's frustrations showed in a third-quarter fight between running back Devonta Freeman and Donald that led to Freeman's ejection. The two locked up after Ryan's pass off Mohamed Sanu's hands was intercepted by Cory Littleton deep in Atlanta territory with the Rams leading 20-3.
Freeman had his hands on Donald's facemask before throwing a punch at the much bigger Donald, which appeared to make little if any contact. Umpire Terry Killens finally pulled Freeman away from Donald, taking Freeman to the ground in an apparent unintentional body slam before quickly helping Freeman to his feet.
Each player drew unnecessary roughness penalties, and Freeman was tossed for throwing the punch.
The turnover left the Rams at the Atlanta 16 for a short drive capped by Goff's 1-yard scoring run.
With running back Ito Smith already out with an injury, Freeman's ejection left Brian Hill and return specialist Kenjon Barner as the options at running back.
Goff completed 22 of 37 passes for 268 yards and two touchdowns. Blake Bortles took over on the Rams' final possession.
Atlanta's offense struggled even before Freeman's exit. The Falcons were limited to 224 yards.
Ramsey, the former All-Pro cornerback, was acquired by Los Angeles from Jacksonville on Tuesday for three draft picks, including two first-rounders.
As expected, Ramsey shadowed Falcons standout Julio Jones much of the game. Jones had six catches for 93 yards in his first matchup with Ramsey.
Ramsey moved into a revamped secondary. The Rams placed cornerback Aqib Talib and safety John Johnson on injured reserve this week and traded cornerback Marcus Peters to Baltimore. That left safety Eric Weddle as the only holdover starter.
The new mix held up well. Ryan completed 16 of 27 passes for only 159 yards with an interception.
Goff threw a perfectly placed 13-yard touchdown pass to Todd Gurley early in the second quarter. He added an 8-yard scoring pass to Gerald Everett in the third quarter.
Everett, an Atlanta native, celebrated by doing the "Dirty Bird" dance made famous in the Falcons' 1998 Super Bowl season.
Smith was carted off the field late in the first quarter with head and neck injuries following a helmet-to-helmet hit from Rams linebacker Cory Littleton. There was no flag. Smith did not return.
The Rams will remain in Atlanta to practice at Georgia Tech for a few days before flying to London for next Sunday's game against Cincinnati.
The Falcons will play the Seahawks in another home game next Sunday. Seattle leads the series 10-8, including two playoff games won by Atlanta.