Jensen's deal, which will take him up to a decade of service for Burnley, includes provision for a testimonial in recognition of his contribution.
The 37-year-old Dane played just a handful of games last season due to injury and the form of Lee Grant, but is in manager Eddie Howe's plans for a further year.
"I'm pleased he has signed," said Howe. "We always wanted him as part of the squad and his experience will be vital. He has a big part to play in being a role model for the other players and the young group of goalkeepers we are bringing through. It's important to have good professionals around them."
Jensen has made 307 appearances for Burnley since signing on a free transfer from West Brom in June, 2003.
Meanwhile, fellow Clarets goalkeeper Jon Stewart, 23, has also signed a new contract. The former Bournemouth goalkeeper provided back-up to Grant and Jensen for much of last season before spending a successful spell on loan with Alfreton.