The Italy international has now sealed a switch to San Siro for the 2014-15 campaign, with the club retaining the option to make the move permanent in future.
Taider, meanwhile, becomes Southampton's fourth signing of the summer, following the arrivals of Ryan Bertrand, Graziano Pelle and Dusan Tadic as Ronald Koeman continues to rebuild his squad in the wake of a number of sales.
Calum Chambers, Adam Lallana, Dejan Lovren, Rickie Lambert and Adam Lallana have all left the club this summer but Koeman is pressing ahead with reshaping his squad, with Taider joining on the back of a successful World Cup with Algeria.
"It's part of our philosophy for the club to bring in talented young players who like to learn and develop themselves, and that's true of Saphir," Koeman told the club's official website. "First of all, he is a talented football player. He's 22 and has already played for Algeria at the World Cup.
"I've spoken to him about the positions he can play in, and it was a good meeting because he showed me real ambition."