The former defender has signed a three-year contract having been given the role of head coach following the dismissal of Michael Laudrup in February.
"We are delighted to confirm Garry as our new first-team manager," Swans chairman Huw Jenkins told swanseacityfc.net.
Monk will be assisted by current coach Josep (Pep) Clotet.
Monk said: "It is the proudest moment of my career. I'm honoured the club thinks so highly of me to give me this chance.
"It's the opportunity of a lifetime - and one I fully intend to take."