Peter Taylor's Eagles had to come back twice to level, when Jensen made two tremendous stops on his line and then four minutes into stoppage time produced a flying save to tip Michael Hughes' 20-yard shot over the crossbar.
Burnley were well stretched in that frantic finish, but Jensen's defenders all played a part in preventing Palace from snatching all three points.
Not that Burnley didn't deserve their point as they a lot slicker than Palace in the first-half and deservedly went in front on 23 minutes.
Former Crewe Alexandra striker Steve Jones cleverly got the better of full-back Danny Butterfield before passing across the edge of the box for Alan Mahon to slot low past Hungarian goalkeeper Gabor Kiraly and into the corner of the net.
The closest Palace came to scoring came after summer recruit James Scowcroft hit the post with a tremendous low drive from 25 yards.
After the break, Palace wasted no time and central defender Leon Cort climbed above the Clarets defenders to nod Mark Kennedy's inswinging free-kick past Jensen four minutes in to level at 1-1.
But Burnley hurt Palace almost straight away, as Kyle Lafferty took advantage of a fortunate deflection which fell straight at his feet and he simply prodded the ball past Kiraly to restore the lead.
The Eagles were not finished though and 16 minutes from the end Jobi McAnuff did the business on the left, before floating over the perfect cross for Scowcroft to head firmly over Jensen and into the far corner of the net.
At the other end, substitute Steve Foster hit the underside of the crossbar before Jensen began performing miracles and left the home side in despair at how they failed to get the three points.