The 34-year old Danish stopper, also known as 'The Beast', has decided to stay with the Clarets for the next two seasons.
His commitment to Turf Moor was praised by Burnley boss Brian Laws who labelled Jensen as a 'great servant'.
"We are delighted Brian has agreed to sign the two-year deal we offered. He is a quality goalkeeper and someone who has served this club with distinction," he told the club's official website.
"He is not only a great servant, but a great ambassador and, even at his age, we feel Brian has good years ahead of him. We hope to drive him on further this summer because clearly we would like to strengthen in that area.
"It is something we are trying our hardest to achieve so that Brian has real strong competition, which can only improve his performances for the club."