Curtis was initially appointed caretaker coach of the club following the departure of Garry Monk on December 11.
"We firmly believe it is the right decision for Swansea City," Swans chairman Huw Jenkins told the club's official website.
"Alan has been with us through good and bad times and was part of the management team that helped the club secure its Football League status over 12 years ago.
"He is fully aware of the next important job he has ahead of him to dig deep and find the required levels of performance and motivation to secure our Premier League status – our main goal this season.
"We believe there is nobody with more knowledge and experience of the club to do that than Alan Curtis.
"He has served Swansea City so well for more than 40 years and played an integral part in our recent success under the likes of Roberto Martinez, Paulo Sousa, Brendan Rodgers, Michael Laudrup and Garry Monk."