"I can announce that we managed to get the services of Benjamin Pavard from July 1, 2019", Salihamidzic told reporters during a Bayern training camp in Qatar, without giving any financial details of the transfer.
The 22-year-old French defender, who can operate at centre-back or right-back, has agreed a five-year deal with the current German Bundesliga champions.
The defender will join Bayern on a five-year deal after they activated his €35m (£31m) release clause.
Pavard helped his country win the 2018 World Cup in Russian Federation and scored goal of the tournament, against Argentina.
His memorable strike against Argentina in the last 16 was also voted Goal of the Tournament, beating the likes of Cristiano Ronaldo and Lionel Messi.
With the transfer window now in full swing, Bayern Munich have confirmed the signing of a French 2018 FIFA World Cup victor, who will join them by the end of the 2018/19 season.
Niko Kovac's side trail Dortmund by six points in the table - the first time in eight seasons that they have not led the way going into the winter break.
He has featured 14 times for a struggling Stuttgart this season, with the club now 16th in the Bundesliga table.