Robbie Blake had put the hosts in front with a superb near-post volley, only to upend Patrice Evra at the other end just before the break.
Michael Carrick took responsibility ahead of Michael Owen. But Jensen denied him, making a fine stop to allow Burnley to celebrate an amazing night.
Burnley rocked badly in the opening stages and Michael Owen harshly had a goal ruled out for offside when he appeared to have timed his run onto Patrice Evra's through-ball to perfection before beating Jensen at the second attempt.
Martin Paterson thought he was going to open the scoring for the Clarets when he sprang the United offside trap, only to see Ben Foster make a brave block, then Carrick followed suit as Blake tried to turn home the rebound.
United could not relieve the pressure though and when former Manchester City full-back Stephen Jordan drifted a cross to the far-post, Blake was on hand to let fly with a superbly-struck volley which was past Foster before the United keeper had a chance to react.
The odds must have been on Owen taking the penalty when Evra was upended by Blake as he nicked the ball towards Burnley's goal.
Instead, Carrick took responsibility - and Jensen emerged the hero, just as he did last season when his spot-kick saves turfed Chelsea out of the Carling Cup.
A long-range effort from Wayne Rooney that sailed harmlessly over was the best United could manage after the break until Ji-Sung Park created a bit of space for himself and forced Jensen into a fine full-length save, with Jordan hammering away the rebound.