"We are delighted Brian has agreed to sign the two-year deal we offered," said manager Brian Laws.
"He is not only a great servant, but a great ambassador and we feel Brian has good years ahead of him."
Jensen's penalty save in the opening home game of the season helped Burnley beat Manchester United.
The Copenhagen-born keeper is Burnley's longest serving professional, having joined from West Bromwich Albion in June 2003.
He is a quality goalkeeper and someone who has served this club very well with distinction Burnley manager Brian Laws The former AZ Alkmaar keeper joins Michael Duff and Steven Thompson in signing new deals.
"He is a quality goalkeeper and someone who has served this club very well with distinction," added Laws.
"We hope to drive him on further this summer because clearly we would like to strengthen in that area.
"It's something we are trying our hardest to achieve so that Brian has real strong competition, which can only improve his performances for the football club."