The 26-year-old winger enjoyed a successful four-and-a-half seasons at the Philips Stadion, where he helped Phillip Cocu's side win back-to-back Eredivisie titles in 2015 and 2016.
Narsingh had been a target for former Swans manager Bob Bradley, and his successor, Clement, is delighted to secure the signature of the Netherlands international on a two-and-a-half year deal.
"He is a player the club had identified before I arrived, but I am more than happy for that to go ahead," the Englishman told the club's official website.
"When I was at Bayern we played against PSV Eindhoven and he played. He scored a wonderful goal in the game in Eindhoven.
"He is going to bring another dynamic to the team. He has pace, he can score goals and has experience of playing at a good level.
"(He's been a) Dutch champion a couple of times and also has experience of Champions League football so we are bringing in a good calibre of player."
Narsingh admits compatriot Leroy Fer played a huge role in his decision to move to the Liberty Stadium.
"A dream has come true playing in the Premier League, playing for a big club like Swansea, it was not difficult to think about it," he said.
"I talked to Leroy before I came here and he told me it is a good club with great team-mates and fantastic supporters. It helped me a lot in making a decision, so it was easy for me to say yes to Swansea.
"This is the best league, and with Swansea playing good football I knew it was a good club for me. I'm 26 now and it's a good age to come to the Premier League. I won two titles in Holland, so it is a great time to come here."