After agreeing a £17million switch to United last month, the 26-year-old Canada-born player brought the curtain down on a decade in Bavaria with a farewell press conference in Passau.
"I would never have thought that I would move to Germany," he said.
"I am very happy that FC Bayern gave me such an opportunity. I wish them all the best. The last 10 years were a dream."
Although Hargreaves was not able to leave the Bavarians on a high note - they finished last season in their lowest Bundesliga position since he joined from the Calgary Foothills club - he still managed to achieve all he set out to do in Germany.
He won four Bundesliga titles with Bayern as well as the 2001 Champions League and Intercontinental Cup (World Club Championship).
Hargreaves is hoping his move to Old Trafford will not interrupt his run of success.
He said: "I set lots of targets over the last decade and I have also fulfilled them. I expect that to continue.
"A new era is starting for me now and I hope to achieve even more. I don't have any concrete aims, but you have got to win everything at Manchester."