Salah celebrated his two-year contract extension with an assist for Luis Diaz's opener, with the strike marking the Egyptian's 45th goal involvement of the Premier League season, breaking the record for a 38-game campaign.
The 33-year-old midfielder, who joined City in 2015, was playing his first home match since it was announced he would be leaving the club at the end of the season
Nico O'Reilly added City's fifth in the 79th minute as the champions provisionally climbed to fourth in the table, two points ahead of fifth-placed Chelsea who have played a game less.
The English top flight needed just one win from its clubs in any of the three European competitions to secure one of two extra berths - on top of the four granted to the top four in the table - awarded to the highest-ranked countries in the UEFA coefficient ranking.
Pep Guardiola's team are fifth in the table on 52 points, one behind Chelsea, and United are languishing in 13th place on 38.
Liverpool remain top with 73 points from 31 games, 11 ahead of Arsenal. They need a maximum of 11 points from their remaining seven matches to win the title.
Second-placed Arsenal have 62 points from 31 games, 11 behind leaders Liverpool, who have a game at hand when they go to Fulham on Sunday. Everton climb to 14th place with 35 points from their 31 matches, 15 points clear of the relegation zone.
Tonali fired Newcastle into fifth place in the Premier League as the Italian midfielder's 74th-minute rocket beat Brentford goalkeeper Mark Flekken from an improbable angle.
Victory means Liverpool need just 13 points from their remaining eight games to be assured of a record-equalling 20th English top-flight title.
City's victory puts them back in pole position to qualify for next season's Champions League, just days before they try to save themselves in this season's competition.
Enzo Maresca's team were just two points behind leaders Liverpool in mid-December, raising hopes of an unlikely title challenge before losing their way.
United crumbled to a sixth loss in 11 Premier League games since Amorim arrived from Sporting Lisbon to replace the sacked Erik ten Hag.
Foden netted either side of Mateo Kovacic's eye-catching goal as City blew Ipswich away with three first-half goals at Portman Road.
Arteta conceded Arsenal were culpable for the defensive mistakes that allowed Villa back into the game.
Mikel Arteta's second-placed side moved within four points of Liverpool thanks to their stirring fightback in Wednesday's 2-1 win against arch rivals Tottenham.
Liverpool's second consecutive league draw opens the door for Arsenal to cut the gap at the top to four points when they host Tottenham in the north London derby on Wednesday.
Chelsea's second successive defeat after their Boxing Day loss against Fulham left them without a win in their last three matches.
Just two months into his tenure, Amorim conceded this week that his job is on the line unless results begin to improve.
Guardiola's struggling team won for the first time in five league games to climb to a provisional fifth in the Premier League, but are still way off leaders Liverpool, who have 11 points and two games in hand over the reigning champions.
