Ferguson bows out next Sunday after securing a 2-1 win over Swansea in his final home game as manager.
After the victory, he addressed the crowd and told them to get behind his replacement. The current Everton boss will start at Old Trafford on July 1.
"I would like to remind you this club stood by me in bad times, the players and the staff," he said.
"Your job now is to stand by the new manager."
Ferguson paid personal tribute to Paul Scholes, who started Sunday's game and will also retire in the summer, and Darren Fletcher, who continues his recovery from a chronic bowel condition.
And he insisted he would still be a regular presence at Old Trafford.
"I will be able to go along and watch them rather than suffer with them," he said.
"Those last-minute goals, the comebacks and even the defeats are part of this great football club. It has been an unbelievable experience."
Ferguson added: "I have been fortunate to manage some of the greatest player in this country, let alone Manchester United.
"They have represented our club in the proper way.''
Ferguson said he had "no speech in mind'' and would just "ramble on'' before admitting at one point that he might start "bubbling''. However, he did not.
"I wish the players every success,'' he said.
"I know how good you are. You know the jersey you are wearing and you know what it means to everyone here.
"Do not let yourselves down.''
Ferguson began his address by thanking all the players, staff and supporters of the club and said: "You have been the most fantastic experience of my life.''
Rio Ferdinand was delighted to give Ferguson a winning send-off in his final Old Trafford match following the "devastating" news that he would be stepping down from his post at the end of the season.
Ferdinand smashed home an 87th-minute goal to earn United victory after Michu had cancelled out Javier Hernandez's first-half opener for the Premier League champions.
"To win at Old Trafford, the man's desire to win football matches, to keep bringing success to the football club, he deserved to have three points in his last home game," said Ferdinand, who revealed the players' shock when Ferguson announced to them he was stepping down.
"Obviously it was a personal moment between him and the squad, it's emotional, he's had so many years here, it's his second home," he told Sky Sports 1.
"For us to hear him say it, we were devastated, but as you know this is a great club, we have to move on and that's what he wants."
Ferdinand is out of contract at the end of the season but the 34-year-old is hoping Ferguson can influence the powers that be at United to give him a new one.
He said: "It'd be nice. I'll have to speak to the board and hopefully the manager puts in a good word in for me!"
United lost the Premier League title in agonising fashion to Manchester City last season but bounced back to reclaim the trophy in convincing style this season - the 13th time under Ferguson.
Part of the reason for this season's success was the arrival last summer of the prolific Robin van Persie from Arsenal.
The Dutchman was thrilled to get his hands on a Premier League winner's medal after failing to manage it during eight seasons with the Gunners.
He said after lifting the trophy: "It was amazing, it was heavy!
"I couldn't have wished for this year to be any better, only in my wildest dreams could I have wished it be like this."
He also hailed Ferguson, adding: "For me it was a massive honour to have worked with him. I want to thank him, he's been unbelievable. He's a great manager and an even greater person. He deserves all the credit. He's amazing."
On finding out Ferguson was leaving, Van Persie added: "It was a really sad day. Everybody understands it."
Meanwhile Scholes received a rapturous reception from the Red Devils fans having helped the club to 11 Premier League titles.
And the midfielder said afterwards: "It's been a good day. We won the game and that's all that matters. We've picked the trophy up and we'll celebrate tonight.
"I haven't played for four or five months so it was just nice to get through the game and win it.
"I thought it went quite well. You thought it would have been quite emotional for him (Ferguson) but he spoke well at the end of the game and he has done a fantastic job here, that goes without saying.
"He's just happy now and is looking forward to what the team are going to do next year and hopefully we can go on from here and win more trophies."
On his retirement, he said: "I know it's the right time. It's not the season I would like to have had but we've won a trophy and that's all that matters to the team.
"It's a great day. We won the game and won the league."
Another of United's veterans, winger Ryan Giggs, said Ferguson was determined to win the match and to do it with a good performance.
"He (Ferguson) just said play the game and not the occasion. He wanted to win the game, he wanted his last game at Old Trafford to be entertaining and to win and to do all the things we've done over the years since he's been manager," said Giggs.
"All players have got a part of play, whether they are players who have not achieved much or older players who drive them on.
On his favourite memory of Ferguson, the 39-year-old Welshman said: "There's not a favourite time. The memories that stick out are ones from when I was younger and 13, 14, playing for Salford boys. I would look over to the sideline and there is the manager of Manchester United. That is amazing for a 13-year-old and the relationship has just built."
While the likes of Scholes and Giggs are celebrating the latest in a long line of title successes with United, defender Phil Jones - like Van Persie - was enjoying life as a first-time winner.
The 21-year-old, who joined United from Blackburn in 2011, told Sky Sports 1: "You dream of things like that as a kid and you hope it one day happens, and it's an incredible feeling.
"It's amazing. The supporters have been fantastic all season, especially after last season where we all know what happened. We've bounced back really well and that's the Man United way."
Ferguson recently suggested Jones had the ability to become the greatest player in United's history.
Responding to that, the England international said: "It makes me feel bigger. I admire the comments but it doesn't stop there, you have to keep playing and as long as I play the way I have I'm sure things will be good."
He added of Ferguson: "He's a fantastic manager, everyone knows that, you can't speak highly enough of him but what he's done for me personally has been fantastic and I can't thank him enough."
Meanwhile Swansea manager Michael Laudrup, who has been linked with the Everton job when Moyes leaves for Old Trafford, insists he wants to remain with the club for their European campaign next season - but hinted that he may consider offers after that.
He said: "Next season I want to try to consolidate Swansea. There is a lot of challenge there, we're playing in Europe and we would like to have that experience as a club, as a team and I would like to be a part of that.
"Then let's see what happens in the future."
He also claimed that claimed that finishing in the top 10 of the league is an even greater achievement that winning the Capital One Cup.
"This season will always be remembered for the cup but to be in the top 10 is something really, really great," Laudrup said.
"Money is not all not in life, and not in football but it's quite important and to have the lowest budget in the league means to finish in the top 10 may be an even bigger result.
"Our aim has to be to try to consolidate in mid-table. That sounds a bit arrogant but we finished 11th last year and eighth or ninth this season. That's what we have to aim for next season as well and it won't be easy as I know some of the clubs below us want to spend heavily so it's another huge challenge."
Laudrup also paid tribute to Ferguson, saying it was "an honour" for him to be at Old Trafford for the United manager's final home match after nearly 27 years in charge.
He added: "It's something very, very special in football. I don't even remember Manchester United with another manager. It's incredible.
"But if you talk about being manager at one club for 27 years - then no thank you!"