La Liga gave Barcelona permission to make a signing outside the transfer window after Ousmane Dembele was ruled out for six months with an injury.
La Liga rules allow clubs to apply to make new signings if a player is ruled out for more than five months.
Braithwaite’s contract will run until the end of June 2024.
His deal also includes a buy-out clause of 300m euros (£251m).
The rules allowing Barcelona to sign the 28-year-old stipulated the player they brought in must be a free agent or already playing in Spain and that they can not play in Uefa competition.
Braithwaite signed a four-year-deal at Leganes in July following a spell on loan from Championship side Middlesbrough last season.
He scored eight goals in 27 games since joining them permanently and has six of Leganes’ 18 league goals so far this season.
Leganes are in the relegation zone of La Liga and have lost the forward, capped 39 times by Denmark, after Barcelona met his buy-out clause.
“We will raise our voice,” said Leganes director general Martin Ortega. “We are in a situation of enormous and serious damage.
“We cannot understand the current regulation that a club, having a long-term injury, can perform this operation unilaterally and transfer their problem to our club.
“We consider that there is a regulation that is unfair, from which Barcelona has benefited. The one who is damaged by this is Leganes.”
‘Natural flair’ and disagreements with Pulis – who is Braithwaite?
According to his official website, Braithwaite “showed a natural flair for the game” as a baby and describes himself as the “fastest player in Denmark”.
Raised in the Danish town of Esbjerg, for whom he began his youth career, he made his senior debut in 2009 and helped the club to the Danish first division title in 2011-12 and cup in 2013.
Despite that success, the striker hinted at feeling undervalued during his time at Esbjerg.
“They may not have known what I could do, but I knew,” he said.
Departing Denmark for Ligue 1 side Toulouse in 2013, Braithwaite made 137 appearances before joining Championship side Middlesbrough in 2017 on a four-year contract.
Initially signed by manager Garry Monk, the forward struggled for opportunities under his replacement Tony Pulis. “Disagreements” between the pair ensued, leading to loan moves to Bordeaux and then Leganes.
He made his Leganes move permanent in the 2019 summer transfer window.