The move ends interest from both Bayer Leverkusen and Bayern Munich, who had also been credited with tracking the left-footed defender before he opted to join RB Leipzig.
Estève arrives in Saxony with considerable experience for a player his age, having previously worked under current Bayern head coach Vincent Kompany at Burnley, including during the club's return to the Premier League last year. Kompany moved on to Bayern and Estève has now completed his own switch to Germany.
RB Leipzig sporting director Marcel Schäfer explained the thinking behind the signing.
"The signing of Maxime is a very important transfer for us. We're looking forward to a centre-back who, despite his young age, has already gathered a lot of experience in France and England and made an impressive development at Burnley. We're convinced that Maxime can help our team immediately and take on an important role for us," Schäfer said.
Estève himself expressed enthusiasm about the move after putting pen to paper.
"I'm very happy to be a player of RB Leipzig from now on. The club stands for intense, brave football, for development and for big ambitions.
"That fits me very well. The conversations with those in charge were very convincing. I felt that the club has a clear plan for me. Now I'm looking forward to getting to know the team and the fans," he said.
The defender's rise has taken him from French football through to England, where he became a key figure in Burnley's squad during a turbulent period that included both relegation and promotion. Leipzig will hope his experience of high-pressure situations translates into an immediate contribution as they look to strengthen their defensive options for the season ahead, with the club having fought off genuine interest from two of the Bundesliga's biggest names to get the deal done.
Estève's arrival represents a significant statement of intent from Leipzig, who moved decisively once it became clear the player was prepared to commit his long-term future to the club rather than wait for a potential move to Bayern or Leverkusen.
His versatility and composure make him a natural fit for Leipzig's possession-based approach, and the seven-year contract underlines the club's ambition to build around him for years to come rather than treat the signing as a short-term fix.