Jensen had been the hero for the Clarets in their quarter-final penalty shootout victory over Chelsea and the big Dane, affectionately known as 'The Beast' was keen to ensure that Owen Coyle's side continued to enjoy their fairytale cup ride.
Jensen made the first of a serious of saves as early as the fourth minute to deny striker Nicklas Bendtner.
The home side went in front with their first real threat on goal when Robbie Blake found Chris Eagles and his ball to the near post was turned in by Kevin McDonald, after goalkeeper Lukasz Fabianski had been put under pressure by Martin Paterson.
Bendtner had an early opportunity to equalise when sent clear by Carlos Vela, but Jensen was equal to the task.
Jensen was then called into action to save from both Mark Randall and Vela, before in a rare Burnley attack Paterson had a left-foot strike at goal, but his effort lacked power.
A lovely backheel by Bendtner sent Randall clean through on goal, but Jensen produced another fine save to deny the young midfield player who had spent a short while of last season on loan at Turf Moor.
Jensen ensured Burnley went in at the interval in front with another remarkable save after Bendtner was again the provider with a pass to Vela.
Fran Merida should have tied things up just two minutes into the second half when he beat three Burnley defenders, but shot wide of the far post.
That was to prove an expensive miss as Arsenal were caught napping by a quick thrown-in from Eagles which sent MacDonald through on goal to score his second of the evening.
The home side should have made the game safe on 64 minutes when Blake chipped the ball to the far post, but Paterson's header was saved by Fabianski.
However, the Clarets defended bravely as the game wore on and this victory, their third against London Premier League opposition sealed a memorable evening.