The scores had been level at 1-1 after 90 minutes and also following 30 further minutes of extra-time.
Chelsea missed two of their spot-kicks with Mikel the fall guy when his penalty was brilliantly saved by Brian Jensen.
Didier Drogba had put the Londoners ahead after 27 minutes only for the Championship side to level the scores through substitute Ade Akinbiyi on 69 minutes.
Neither side could fashion a winner in the remaining time and it was Chelsea that cracked under the pressure of spot-kicks to send the 6,000 Burnley fans into delirious rapture
Chelsea claimed they were taking the game seriously against lower league opposition, but they could not summon the energy or will to record the match for their usual match-day outlets.
They also made six changes from their 2-0 victory over Blackburn Rovers on Sunday.
Chelsea were forced to replace the injured Juliano Belletti with Frank Lampard on 25 minutes.
Within seconds of the change Chelsea took the lead when Lampard passed to Drogba who danced through the Burnley rearguard on the left of the area, before planting an excellent curling right-footer around Jensen and into the left-hand corner of the net.
Chelsea rattled the crossbar when Branislav Ivanovic's ducking header from six yards out from Florent Malouda's byline free-kick rebounded off the woodwork.
The second half started much as the first with wayward shooting but gradually Chelsea took control and Ivanovic was unlucky with a powerful header on 55 minutes which just went over the Burnley goal.
Owen Coyle, the visitor's manager, brought on Akinbiyi on 60 minutes for Martin Paterson in an attempt to rough up the central pairing of Alex and Ivanovic. Chelsea boss Luiz Felipe Scolari replaced Drogba with Franco di Santo eight minutes later.
But within seconds Carlo Cudicini fumbled a cross by Chris Eagles on the left and Akinbiyi tucked home from ten yards to level the tie.
Di Santo almost put the Londoners back in front with four minutes remaining but his downward header from six yards from a Lampard cross was held on the line by Jensen.
But Di Santo missed a sitter on 90 minutes when he blasted wide from ten yards out when free on goal to put the tie into extra time.
Lampard had the ball in the net but was judged to be offside three minutes in and Chelsea went down to ten men when Di Santo hobbled off the field injured with a torn hamstring.
From a corner on 101 minutes, Lampard almost scored directly but the ball hit the nearest upright and ricocheted clear.
Akinbiyi's diving header from just inside the Chelsea area was well held by Cudicini as Burnley forced the pace with the extra man.
Referee Stroud then brandished a second yellow and then red card for Steven Caldwell for a foul on Malouda with four minutes remaining. But the game drifted into the tension and ultimately victorious penalties for the visitors.