Though a transfer fee was not officially revealed, it is estimated that the Bavarians met his €2.8 million [£2.5m] release clause.
"I'll never forget May 19," the 22-year old striker said. "It makes me proud that Bayern Munich are my new employers. I am really looking forward to it and I have big plans."
The forward will wear the squad number nine, previously worn by Gerd Muller and Giovane Elber at the club.
He added: "I was surprised by that [wearing No.9], but I'm very proud. Nine is the biggest number for a striker. These are big footsteps, but they're to be followed in."
The striker signs following an excellent season at Cottbus, in which he scored a 2. Bundesliga-leading 25 goals and provided seven assists. He will not be the last player the Bavarians sign during the transfer window.
"I think that within the next two or three weeks we will announce one transfer or another. Until then, we have a little more work to do," sporting director Christian Nerlinger said.
"We will reinforce certain areas so we can win the German championship and also contend again in Europe," Nerlinger said, before specifying his top priority: "It is clear that we want to strengthen ourselves in defensive areas."
Bayern are said to be close to completing a deal for Schalke goalkeeper Manuel Neuer and the signing may be completed shortly after die Knappen's last match of the season, Saturday's DFB Pokal final.