The fourth round tie finished 1-1 after normal time and neither side could add to that during extra time so penalties were required.
Burnley had the chance to win it after five penalties due to Wayne Bridge's miss but Wade Elliot blasted his shot wide.
Michael Duff then scored the first sudden death penalty for the visitors and Jensen made the crucial save from John Obi Mikel to send Owen Coyle's side into the quarter-finals.
Chelsea had dominated proceedings in the first half with Bridge shooting over the bar and Salomon Kalou being denied by Jensen on the edge of his box, before they managed to get the breakthrough.
With 30 minutes gone Didier Drogba was on hand to put the blues in front. Lampard picked him out inside the ox and the Ivorian's close control allowed him to get the shot away. The goal was Drogba's first of the season on his return from injury.Just before half time the Blues should have doubled their lead but Branislav Ivanovic could only head Florent Malouda's cross onto the woodwork.
With an hour gone the Blues took off Drogba and replaced him with Franco di Santo and were made to regret the decision soon after.
Burnley substitute Ade Akinbiyi made it 1-1 and got his side back in the match when he poked home a rebound when Carlo Cudicini failed to hold on to the ball.
Chris Eagles was sent though on goal and although Cudicini saved one-on-one with the midfielder the ball broke to Akinbiyi who tucked it home.
With no further addition to the scoreline extra time was called for and no sooner had that started than the Blues suffered another setback.
Di Santo was forced to leave the game with a hamstring injury but could not be replaced as Chelsea had used all three of their substitutes earlier, thus forcing them to go down to ten men.Both teams could have won it during the extra period, Lampard shot over the bar and Cudicini made a smart save to deny Alan Mahon. The best chance fell to Alex but the Brazilian defender blazed high over the bar from four yards out.
Burnley captain Steven Caldwell was then sent off for a second bookable offence meaning it was ten against ten heading into the shootout.
The Burnley players held their nerve well and looked to be heading into the last eight when Jensen saved Bridge's penalty but Elliot sent their fifth penalty into the stand to keep the match alive.Malouda converted his spot kick to force sudden death and it seemed as if the Premier League side would not let it slip from their grasp again.
But Duff scored his spot kick and Jensen made another fine save to deny Mikel and send the Championship side through and send the Premier League leaders out.