Rodgers put the finishing touches to the deal, having previously agreed in principle to stay on.
The 39-year-old said: "I'm delighted to sign the new deal. I hope the club and supporters feel it is still a good fit for everyone. I've been very pleased with my work over the last 19 months. I can only promise that over the next three and a half years I will continue to fight for the club and its people to help us become a sustainable Barclays Premier League club."
He added: "I very much appreciate what Swansea as a city, a club and supporters have given me and my family over the last 19 months, so signing the new contract was one of the easiest decisions of my life."
The Northern Irishman moved to the Liberty Stadium in July 2010 and guided the Welsh club to the top flight last year.
Since then he has overseen an impressive start to life in the Premier League as Swansea stand 10th in the table, 10 points clear of the relegation zone.
Swans chairman Huw Jenkins said: "It's very good news. We are under no illusions about the fact that we have got one of the best young managers around working for our club.
"Brendan has shown what he is capable of over the last couple of seasons and what he has done for us has been very pleasing.
"He is fully behind and supports the philosophy of the club and how we want to work. His personality and immense qualities highlight the importance of finding the right person to work with us to take the club forward.
"He is very important to us and that's not going to change, so it was very, very important to get the contract sorted out."