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 said: "I would like to remind you this club stood by me in bad times, the players and the staff. 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.
"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".
"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."
Source: PA
Source: PA