Rodgers, who has also had a spell in charge of Watford, has been out of work since leaving Reading by mutual consent back in December.
But the 37-year-old will now take up his new role at the Liberty Stadium with immediate affect.
Rodgers becomes the Swans' third manager in as many seasons and replaces Paulo Sousa, who left the club to join Leicester earlier this summer.
Rodgers told www.swanseacity.net: "It's a very proud moment for me. I'm very proud to become manager of Swansea City, a progressive club that plays the game the right way.
"My aim is continue that philosophy and help the club progress. I want to play offensive, attacking football, but with discipline.
"I will inherit a wonderful group of players and if I can help them develop and improve I will be very pleased.
"I've seen the passion of the Swansea fans for myself when I was here last season and I can assure them I've come here to be a winner. I'm really looking forward to the challenge."
Swans chairman Huw Jenkins added: "I'm delighted with the appointment. It's taken us a bit of time, but I'm sure Brendan will prove well worth the wait."
Source: PA
Source: PA