Kean, 43, took over in a caretaker capacity and was initially handed the manager's role until the end of the season.
"I am delighted to commit my future to Rovers at the start of what I see as a massively exciting new era for the club," he told Blackburn's website on Thursday.
"The owners have shown great faith in me and I will do all I can to repay them. This is a great opportunity for me and I intend to grasp it with both hands."
Kean previously worked as Chris Coleman's assistant at Fulham, Coventry City and at Real Sociedad in Spain after a journeyman playing career that included spells at Celtic and at Portuguese side Academica.
POSITIVE IMPRESSION
Blackburn, who were taken over by Indian poultry business Venky's in November, said last month they wanted an experienced manager and Diego Maradona was touted as a possible long-term replacement for Allardyce.
Kean, however, has left a positive impression by winning three of his opening seven games including a 3-1 victory over Liverpool at Ewood Park.
"We have great confidence in Steve, someone who impresses us with his work ethic, enthusiasm, positivity and football philosophy," said Venky's chairperson Anuradha Desai.
"We feel the playing squad is in very safe and capable hands and we look forward to working with him. Stability is very important to us and this contract is further evidence of our desire to achieve such stability at the club."
Blackburn, who are at home to West Bromwich Albion on Sunday, are 11th in the league with 28 points.
