The Championship club was seeking a replacement for Graham Potter, who left to take over Premier League side Brighton and Hove Albion last month after just one season at the Liberty Stadium.
Welshman Cooper, 39, has been named Potter's successor in what will be his first senior role after stints with the academies at Wrexham and Liverpool preceded a post working with England's youth teams.
He was in charge of the England Under-17 side that beat Spain in the 2017 World Cup final, with Manchester City's Phil Foden, Chelsea's Callum Hudson-Odoi and Borussia Dortmund's Jadon Sancho all starring in that tournament.
"This opportunity is something I've been working towards my whole life, certainly since I started coaching 22 years ago," Cooper said in quotes published on the club's official website.
"I'm extremely grateful to Swansea for giving me this chance. I'm really excited about the challenge ahead."