"What we built together here gives me peace of mind for my new challenge. Everyone one of you has made an impact on me. I appreciate your support and acceptance for who I am.
"The most important thing is the mentality that we have built. My dream is to always have a winning spirit. I wanted nothing more than to help as much as I could and I am proud to have been a part of this group."
Seedorf individually embraced each member of Botafogo's squad, with several of his former teammates also being moved to tears.
"We learnt a lot from him," goalkeeper Jefferson said. "I grew as a man and as a footballer. I hope his departure makes us realize the importance of having an example like that instigating and providing strength to the group.
"We wish him great success and hope that he shines as a coach, just like he did as a player."
Reports in Italy said Seedorf would arrive in Milan on Thursday.
The Rossoneri are currently 11th in the Italian Serie A standings, six points from the relegation zone.
Seedorf has long spoken of his desire to coach and is currently in the final phase of securing his UEFA coaching badges.
Real Madrid coach Carlo Ancelotti praised Milan's decision to appoint the Suriname-born Dutchman.
"I coached him," Ancelotti said. "He's got a strong personality and he will be capable of handling this new experience.
"He knows the club well and I think Milan has taken this decision in the same manner as it has with (Arrigo) Sacchi or (Fabio) Capello. It's always turned out well for them. I wish the best of luck to Seedorf and Milan."
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