Rio Ferdinand slammed in an 87th-minute volley to seal United's 2-1 victory, before Ferguson joined his players in lifting the Premier League trophy for a 13th and final time.
"It's fantastic," said Ferguson, whose haul of 38 trophies at United makes him Britain's most successful football manager. "What a send-off. I was bubbling there for a bit. The atmosphere was unbelievable."
Ferguson's replacement at United, David Moyes, was also feted by fans in his last home match as manager of Everton, with his side beating West Ham 2-0 thanks to a double from Kevin Mirallas to guarantee a sixth-place finish.
Tottenham climbed provisionally to fourth — two points above Arsenal — courtesy of Emmanuel Adebayor's late winner in a 2-1 victory at Stoke, while Newcastle and Norwich secured their Premier League survival with victories over Queens Park Rangers and West Bromwich Albion, respectively.
Sunderland could still go down after conceding an equalizer to draw 1-1 at home to Southampton, leaving Paolo Di Canio's team four points in front of third-from-bottom Wigan ahead of the newly crowned FA Cup winners' visit to Arsenal on Tuesday in their game in hand. Realistically, the three teams still battling against the drop are Sunderland, Aston Villa and Wigan.
Daniel Sturridge scored a hat trick as Liverpool came from behind to defeat Fulham 3-1 in the day's other match.
Ferguson's last home match as United manager was always going to steal the script, however, and the game was given even more prestige as it was the final appearance by Paul Scholes at Old Trafford before the midfielder headed back into retirement.
Ferguson, who dropped Wayne Rooney from the starting lineup and later confirmed the club had rejected the striker's transfer request, started Scholes in central midfield and brought him off after 66 minutes to a standing ovation.
"He is unbelievable, one of the greatest players this club has had or ever will have," Ferguson said of Scholes, who signed professional terms with United in 1993.
With the game approaching what is widely known in British football as "Fergie Time" — coined due to the amount of injury-time goals scored by United in Ferguson's tenure — a corner from Robin van Persie crept through to Ferdinand, who volleyed home high into the net.
Ferguson jumped out of his seat and punched the air in delight, before making his way to the centre circle after full time to deliver an emotional, five-minute speech in the rain.
"If you think about it, those last-minute goals, the comebacks, even the defeats, are all part of this great football club of ours," Ferguson said. "It's been an unbelievable experience for all of us."
Javier Hernandez put United in front in the 39th before Michu equalized for Swansea 10 minutes later. The match was effectively meaningless as United had already wrapped up its 13th Premier League title and is now 13 points ahead of second-place Manchester City, which plays its game in hand at Reading on Tuesday.
The race for the final two Champions League qualification berths behind City and United is still on, though, with Spurs piling the pressure on Arsenal thanks to a come-from-behind victory over a Stoke side that played almost the entire second half with 10 men after Charlie Adam's sending-off for two yellow cards.
Steven Nzonzi gave Stoke the lead before Clint Dempsey equalized in the 20th and then set up Adebayor for the winner in the 82nd.
"This puts more pressure on Arsenal but they are able to cope with it," Spurs manager Andre Villas-Boas said. "We hope Wigan have their day (on Tuesday)."
Hatem Ben Arfa and Yoan Gouffran scored for Newcastle as it came from behind to beat QPR 2-1, while goals by Robert Snodgrass, Grant Holt and Jonny Howson — along with an own goal by Gareth McAuley — helped Norwich to a 4-0 win over West Brom.
In the League Championship, Watford will play Crystal Palace or Brighton in the promotion playoff final on May 27 at Wembley after a thrilling 3-1 win over Leicester for a 3-2 aggregate win.
Manuel Almunia saved Anthony Knockaert's penalty in the fifth minute of second-half stoppage time and Knockaert's shot off the rebound, allowing Troy Deeney to score the decisive goal for Watford about 30 seconds later.