The 22-year-old finalised a switch to the Allianz Arena on Wednesday that could be worth up to €47.5 million depending on bonuses.
Tolisso helped Lyon to the Europa League semi-finals last season and he is brimming with ambition following his move to the German top flight.
"I am delighted to be here at Bayern and to have signed for the best team in Europe," Tolisso told Bayern TV.
"I am very proud to have joined this club, as it has always been one of my biggest goals to join a side like this and that has come true now.
"I hope that I can achieve great things here and help the team win lots more titles. I will do all I can to win silverware with the club. Bayern are one of the biggest clubs in Europe, if not the best.
"Bayern have already won so many titles, both domestically and internationally. They want to win every title and that is also my goal. Unfortunately, I did not win anything with Lyon, so for that reason I am hopeful that I can celebrate winning a few titles with Bayern.
"There was interest from other clubs, but I was determined to come here. The goals that were presented to my advisors convinced me immediately. Right from the start, I really wanted to come here, to become a better player and to win titles."
Tolisso joins compatriots Kingsley Coman and Franck Ribery at the Allianz Arena, while former France international Willy Sagnol has been snapped up to assist head coach Carlo Ancelotti, helping the midfielder feel at home in Munich.
"I spoke to Willy Sagnol on the phone before I signed," Tolisso said.
"I am sure the other Frenchmen at the club will help me integrate. That will be very important for me. I will try to learn the language as quickly as possible in order to properly integrate into the team."