Since returning from a thumb injury that caused him to miss 1 1/2 games, Stafford has eight TD passes and two interceptions in the Rams’ last three games.
Stafford — who was 22 of 37 for 279 yards — connected with Nacua, Demarcus Robinson and Cooper Kupp for scores. Nacua, who left the game briefly with a rib injury, finished with four receptions for 105 yards, including a 70-yard TD in the first quarter. He also had two rushes for 34 yards.
Kyren Williams had 21 carries for 88 yards, including a 1-yard TD late in the fourth quarter to put it out of reach.
Aaron Donald and rookie Kobie Turner combined to sack Joe Flacco for a safety with 36 seconds to play.
Flacco, who became Cleveland’s fourth starting quarterback this season, was 23 of 44 for 254 yards with two touchdowns and an interception in his debut for the Browns (7-5), who have dropped two straight but remain in the thick of the AFC playoff race.
The 38-year-old Flacco was signed on Nov. 20 to the practice squad and elevated to the active roster this week after Dorian Thompson-Robinson suffered a concussion in last Sunday’s loss at Denver.
The Browns trailed 20-19 and had the ball with under seven minutes remaining when Flacco threw an ill-fated bomb that was intercepted by John Johnson III. The Rams safety, who was with the Browns for two seasons, returned it 42 yards to the Cleveland 24-yard line.
Five plays later, Kupp scored his first touchdown since Oct. 15 on a 3-yard pass to make it 27-19.
Cleveland took the opening kickoff and went 75 yards in nine plays, culminating in Flacco hitting a wide-open Jerome Ford for a 24-yard touchdown. Flacco was 4 of 5 for 65 yards on the drive.
The Rams then scored on their next two drives.
Lucas Havrisik made a 44-yard field goal to give Los Angeles points on its opening possession for the eighth time this season.
After Cleveland went three-and-out, the Rams took the lead. Stafford eluded pressure and delivered a perfect pass to Nacua, who caught it in stride at the Cleveland 49 and outraced Kahlef Hailassie and Sione Takitaki to the end zone with 3:59 remaining in the first quarter for his fourth touchdown of the season.
The teams traded field goals in the final 2:06 of the second quarter. Cleveland’s Dustin Hopkins tied it when he was good from 40 yards before Havrisik split the uprights from 28 yards on the last play of the first half to make it 13-10.
Hopkins’ 24-yard field goal capped a 15-play drive in the third quarter before the Rams regained the lead on Demarcus Robinson’s first touchdown with the team. Robinson eluded coverage and caught a 7-yard pass from Stafford with 39 seconds remaining in the period.
Harrison Bryant took a dump-off pass from Flacco and scored from 8 yards out to get Cleveland within 20-19 with 8:56 remaining in the fourth, but Hopkins, who kicked at SoFi Stadium for parts of two seasons with the Los Angeles Chargers, was wide right on the extra point.
INJURIES
Browns: WR Amari Cooper suffered a concussion during the second quarter and did not return. ... Besides Thompson-Robinson, CB Denzel Ward (shoulder) and WR Marquise Goodwin (concussion) were inactive due to injuries.
Rams: TE Tyler Higbee was evaluated for a concussion during the third quarter. ... DB Quentin Lake (hamstring) was inactive.
SUNDAY’S GAME
Jaguars at Browns, 1 p.m., CBS, 1290, 95.7