Ndiaye initially joined Bayern on loan in January and made four Bundesliga appearances under Vincent Kompany last season after making his debut against St Pauli.
His rapid progress also earned him a place in Senegal's 2026 World Cup squad. He made one appearance, coming off the bench in their 3-2 extra-time defeat by Belgium in the round of 32.
The permanent transfer was delayed by uncertainty over Ndiaye's representation and competing claims from Senegalese clubs regarding his development.
Bayern have now confirmed his full acquisition, however, with their existing partnership with Gambinos Stars helping facilitate the move.
Bayern sporting director Christoph Freund praised the youngster's rapid adaptation.
"Bara is a great footballer, who's found his feet in a completely new environment at FC Bayern in a very short space of time and earned the acceptance of the team very quickly," said Freund.
"As a central midfielder, Bara relies on his dynamism and speed on the ball, he's an intelligent player and also brilliant off the pitch. He already speaks decent German and has integrated into our team within a very short time."
Ndiaye is expected to provide midfield depth behind Joshua Kimmich, Aleksandar Pavlovic and Tom Bischof, although Kompany has also tested him on the left wing during pre-season.
The teenager said completing the move was a dream come true.
"Today is a very big day for me and all my family. I want to learn every day from all these great players here at FC Bayern and try to help the team," said Ndiaye.
"It's a dream come true for me, I'm very happy and I also want to be an example to other young players, like my team-mates at Gambinos Stars, that you can make it to one of the biggest teams in the world."
Ivory Coast defender Ousmane Diomande has fulfilled his dream of playing in the Premier League after joining Nottingham Forest from Portuguese giants Sporting Lisbon for a reported 34 million pounds.
Omar Marmoush has been told that the decision over his Manchester City future ultimately rests with him after new manager Enzo Maresca acknowledged the Egyptian forward's desire for more regular first-team football.
Sead Kolasinac experienced a nostalgic return to the VELTINS-Arena on Saturday, lining up for Atalanta against his former club Schalke 04 in a friendly.
RB Leipzig captain David Raum is set to stay with the club through at least the 2026/27 season, though his long-term future remains uncertain.
