This was granted by the Spanish Football Federation (RFEF) on Monday and the club moved quickly to activate Braithwaite's €18m release clause, with the signing announced on Thursday morning by Barcelona's official website.
The Danish global has inked a four-year deal at Camp Nou, after the Catalan giants were handed special dispensation to sign him outside of the transfer window.
Barca decided not to sign a forward in the January transfer window after Luis Suarez was ruled out for four months following knee surgery but need another striker after French global Ousmane Dembele ruptured his hamstring.
La Liga exceptionally allows clubs to make signings if a player is out due to an injury for at least five months.
Braithwaite's contract will run until the end of June 2024. But we don't think the rules are fair. "We will try everything to be able to do this", Leganes sporting director, Martin Ortega, said. We are in a situation of enormous and serious damage.
"We can not hold it against them as they used the rules which now exist".
Braithwaite could make his debut when Barca play Eibar at Camp Nou in the league on Saturday but the rule states he will not be allowed to play in the Champions League.
Braithwaite first joined Leganes in January 2019 on loan from English side Middlesbrough before making the move permanent during the summer for €5m.
The La Liga champions paid the £15m buyout clause for the Denmark worldwide, who scored eight goals in 20 matches for Leganes this season.
The 2018 FIFA World Cup victor, Dembele, 22, underwent surgery in Finland for "a ruptured tendon in his proximal hamstring in his right thigh".