Monk was made interim player-manager in February following the sacking of Michael Laudrup and has since guided the club to four wins and three draws from 13 Premier League fixtures.
The former captain, who joined Swansea as a League Two club 10 years ago, secured their top-flight status after a 4-1 victory over Aston Villa last month, claiming back-to-back wins for the first time in 16 months.
Current coach Josep (Pep) Clotet has been named assistant to the 35-year-old, who said he was thrilled to be given "the opportunity of a lifetime".
"It is the proudest moment of my career," Monk told the club's website.
"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.
"I have had a taste of the job for the last three months and it's something I've really enjoyed. Now I'm really looking forward to the next phase in my life and the club's progression.
"I'll be looking to stamp my authority on the squad and the club - and take it forward. I have a plan in my mind and with the help of the chairman and the board how to implement it."
"We are delighted to confirm Garry as our new first-team manager," chairman Huw Jenkins said.
"As a board of directors, we carefully considered our next step because it was a very important decision for our football club.
"But having gone through that process, we unanimously agreed that the timing was right for Garry to be offered the job on a permanent basis.
"As we look forward to our fourth season in the Premier League, we all agreed that we needed to get back to basics and reinforce the principles that have brought us success in recent years.
"Garry represents all the strong values we hold so dearly as a football club and everyone is looking forward to working with him.
"We have all been impressed with his work-rate and commitment since taking over the reins in February and how he adapted in very difficult circumstances to secure the points we needed to remain in the top-flight.
"Garry will work with the current staff to refocus and improve the playing squad over the summer in order to build a strong, competitive team for another big challenge that lies ahead next season.''