The Gunners and Red Devils had both been linked with moves for the 22-year-old left-back, who has instead opted to remain in France.
PSG officially announced the signing on Thursday.
"I am very happy to have joined a club, and a project, as ambitious as Paris Saint-Germain," Kurzawa said in a statement.
"I will continue to progress alongside some of the best players in the world. Signing for the capital club marks a new step in my career."