Christian Kalvenes timed his first Clarets goal to perfection with an 85th-minute winner to plunge the Seasiders deeper into the Championship relegation mire.
The Norwegian left-back swept a low left-foot drive past Seasiders goalkeeper Paul Rachubka after good work from Chris Eagles and substitute Wade Elliott to settle a miserable and at times farcical encounter ruined by gale force winds.
Yet goalkeeper Jensen, who has enjoyed an outstanding campaign, and leading scorer Paterson both sustained dead legs at Bloomfield Road and are racing against time to be fit for the fifth-round trip to the Emirates on Sunday.
Manager Owen Coyle said: "Brian has got a dead leg, as has Martin Paterson, and that's why I took him off in the second half.
"There are a number of concerns but we'll see how everyone is ahead of Sunday's game."
The fierce gusts which swept a bitterly cold Bloomfield Road meant precious little football was played and the ball would not lie still when Jensen and opposite number Rachubka tried to take dead-ball kicks in the first half.
Referee Lee Mason consulted Coyle and Blackpool caretaker boss Tony Parkes in the 21st minute but a joint decision was taken to continue playing.
Coyle added: "I think the referee wanted to know from the safety officer about our fans' safety, because they were over there standing in the rain.
"There was no cover for them - in this day and age - but we were delighted to reward them with the three points.
"It was a must-win for us tonight and the players I brought back into the side - Kevin McDonald, Clarke Carlisle, Christian Kalvenes and Chris Eagles - all played their part.
"This wasn't a typical Burnley performance, but it couldn't be in those conditions.
"But it was a really brave performance, full of courage and desire and it keeps us up there."
Blackpool may wish the game was abandoned as they went on to lose their fourth home match in succession to slide into deeper trouble.
Parkes accused Mason of trying to get the game abandoned, saying: "It could have gone either way in those conditions.
"We battled away well and I thought we should have had something from the game.
"We should have had a penalty in the first half but the referee was that concerned with abandoning the game that he lost concentration and missed things.
"He kept coming over to the sidelines to ask myself and Owen Coyle to agree with him to abandon the game.
"I don't think he had his eye on the ball all the time and that cost us a win, never mind a point.
"The conditions were okay and it's very rare you abandon a game because of wind and I thought it was very strange the referee wanted to do that."