Burnley were lucky to get away with a handball by the Danish keeper who appeared to use his gloves just outside the area while trying to prevent a Drogba shot.
Defender Juliano Belletti was substituted for Frank Lampard after picking up a knock, and the England midfielder's immediate impact was emphatic, as he played in Drogba with an exquisite pass.
The former Marseille striker left two defenders in his wake before an accomplished finish gave his side the lead and left Jensen with no chance.
It was Drogba's first goal for Chelsea since the Champions League semi-final against Liverpool in April, and the first goal Burnley have conceded in this season's competition.
Malouda nearly doubled Chelsea's advantage after being played in by Kalou, but his shot was pushed away by Jensen.
The French winger then set up Branislav Ivanovic from a free-kick, but the Serbia international's header hit the crossbar and was cleared.
The 6,000 travelling fans were making a lot of noise at Stamford Bridge and were bouyed in the second-half when a nervous-looking Carlo Cudicini struggled to claim a corner, but the danger was cleared.
The Premier League leaders were struggling to exert their dominance on the game amid a spirited Burnley fightback, and appeared happy to slow the game down whenever they reclaimed possession.
Jensen was Burnley's match-winner on a dramatic evening
Burnley's impressive second-half performance then got the equaliser it merited. Winger Chris Eagles was put through on goal and when Cudicini blocked the loose ball, Akinbiyi swept up to equalise and send the Burnley fans wild.
Both teams were smelling victory and the game became stretched as the two sets of players searched for a late winner.
Young Argentinian striker Franco Di Santo almost claimed it in the closing stages of normal time but could only direct his shot narrowly wide of Jensen's post.
Lampard did have the ball in the back of the net in the first period of extra-time after Kalou put him through, but the England man was adjudged to be offside.
A foul on Malouda late-on saw Burnley captain Steven Caldwell receive his second booking, while both Alex and Kalou missed gilt-edged chances to win the game in the closing stages.
The sudden-death defeat means that the Blues have now lost their last five penalty shoot-outs.
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