The Bundesliga champions are believed to have paid Gambinos Stars, the Gambian club that developed the 18-year-old, a fee in the region of €3-4 million.
Head coach Vincent Kompany pushed internally for the club to make the move permanent after Ndiaye impressed since arriving on loan in January, becoming the latest fruit of Bayern's long-running partnership with the Gambian academy.
"Today is a very big day for me and my whole family," Ndiaye said. "I want to learn from all these great players here at Bayern every day and try to help the team.
"For me, a dream came true — I am very happy, and I also want to be an example for other young players, like my teammates at Gambinos Stars, that you can make it to one of the biggest teams in the world."
Bayern's sporting director, Christoph Freund, praised how quickly the teenager had settled into life in Bavaria.
"Bara is a great character who has found his feet in a completely new environment at Bayern Munich in a very short space of time and quickly earned real acceptance within the team," Freund said. "We watched him for a long time, he was captain of his team, and he has made an impressive development over the past 24 months, which he has continued here in Munich."
Ndiaye first moved to Munich in January as part of the partnership between Bayern and Gambinos Stars, an arrangement in place since 2023 that gives the German club close ties to the Gambian side.
His permanent transfer marks the first time that cooperation has borne fruit at senior professional level. He made four Bundesliga appearances for Bayern in the second half of last season, starting twice under Kompany, and is not expected to be loaned out again for the time being.
Ndiaye also featured for Senegal at this year's World Cup, appearing in their round-of-32 exit to Belgium after making his senior international debut earlier in the year.
He stands 1.80m tall and operates as a central midfielder, with Bayern viewing him as eventual competition in the role vacated by Leon Goretzka's departure. His path to Munich began with a two-week visit in 2024, when Gambinos Stars played a friendly against Bayern's Under-19 side, before he spent time training with the club's reserve and youth teams the following year.