Allen passed for 342 yards and three touchdowns and rushed for 82 yards and three more scores for the Bills (10-3), who hadn't lost since Oct. 6 in Houston. He became the first player in NFL history with three passing TDs and three rushing TDs in a game.
Allen capped his third long touchdown drive of the fourth quarter by scoring on a 1-yard dive with 1:00 left.
But Ronnie Rivers recovered the onside kick and Los Angeles (7-6) ran out the clock on its first win over Buffalo since 2012 and its first at home since 1983.
The Rams and Bills combined for 902 yards of offense without a turnover, while Buffalo's 42 points were the most scored in a loss in franchise history. A team scored at least 42 points and lost for just the 22nd time in NFL history.
Hunter Long returned a blocked punt 22 yards for a touchdown for the Rams, who have won six of eight after this improbable outing to stay in the NFC West race.
Nacua had 12 catches for 162 yards and also ran for a score, while Kyren Williams rushed for two TDs as the Rams never trailed while moving above .500 for the first time this season.
Los Angeles led 38-21 entering the fourth quarter before Allen stirred Buffalo from its post-division title-clinching hangover. But after Mack Hollins' 21-yard TD catch with 8:49 to play, Stafford led Los Angeles on a clock-consuming drive that included a gritty 11-yard pass to Tutu Atwell on fourth-and-5 near midfield with less than four minutes left.
After Nacua's catch-and-run TD on a pick play, Buffalo drew a 34-yard pass interference penalty against Quentin Lake on fourth-and-15 from midfield on a drive ending in Allen's third TD rush - but the Bills ran out of time.
One week after Buffalo clinched the AFC East title by trouncing San Francisco in the snow, the Bills started dismally at SoFi Stadium, giving up 24 points in the first half and falling behind by 17 on Cooper Kupp's 17-yard TD catch with 18 seconds left in the third quarter.
Until the frantic fourth quarter, the up-and-down Rams dominated the powerhouse Bills.
Los Angeles had its highest-scoring game since Christmas 2022, and those 44 points were the most allowed by Buffalo all season. The Rams went at least 65 yards on all five of their full drives in the first three quarters while converting on 10 of 11 third downs.
Los Angeles went up 17-7 early in the second quarter when Jacob Hummel blocked Sam Martin's punt. Long scooped and scored the first NFL touchdown for the tight end acquired from Miami last year in the Rams' cost-cutting trade of All-Pro cornerback Jalen Ramsey.
The Rams hadn't blocked a punt since 2018, while the Bills hadn't given up a blocked punt TD since the 2021 season opener against Pittsburgh.
Allen had his second TD run 33 seconds into the fourth.
Up next, the Rams travel to San Francisco to face the 49ers for Thursday Night Football.