The retiring Scot says those associated with the club must back the new manager, just as they did for him during a trophyless spell early in his Old Trafford career
The Scot, who announced his decision to step down as boss after more than 26 years in charge on Wednesday, received a guard of honour before the match against Swansea, as well as a rapturous reception from the home support.
And Sir Alex, addressing the supporters after Rio Ferdinand wrapped up a 2-1 victory with a late strike, appealed to them to back Moyes in the same way that the club backed him when he endured a trophyless spell between 1986 and 1990.
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Moyes will take over from Sir Alex on July 1, with the latter stepping upstairs to take up a seat on the board of directors, and the Scot joked that he will now finally be able to enjoy watching United play.
"This is a thank you to Manchester United," Sir Alex started. "Not just the directors, not just the medical staff, the coaching staff, the players and supporters. It's all of you. This has been the most fantastic experience of my life. Thank you.
"My retirement does mean the end of my life with the club. I'll now be able to enjoy watching them, rather than suffer with them. But, if you think about it, those last-minute goals, the comebacks, even the defeats are all part of this great football club of ours ... so thank you for that."
The 2012-13 Premier League title is Sir Alex's 38th during his Old Trafford tenure, with 209 players having taken the field during that time.
And the Scot addressed the current crop of United talent, thanking them personally for a securing the championship in "fantastic fashion", before highlighting the achievements of the retiring Paul Scholes and absent Darren Fletcher.
He added: "I have been very fortunate that I've been able to manage to some of the greatest players in the country, let alone Manchester United. And all these players today have represented this club the proper way. They won the championship in fantastic fashion. Well done to the players.
"Before I start blubbing, I just want to speak to Paul Scholes, who retires today. He is unbelievable. One of the greatest players we've ever had and ever will ever. I also want to wish Darren Fletcher a speedy recovery and return to the club.
"I wish the players every success in the future You know how good you are, you know the jersey you're wearing and you know what it means to everyone here [at Old Trafford]. Don't ever let yourselves down. The expectation is always there."