Drogba stepped up to rifle home the winning penalty for Chelsea after earlier saving his side with an equalising goal two minutes from time to cancel out what looked like the winner from Thomas Müller as the final finished 1-1.
It was an agonising defeat for Bayern, who saw midfield maestro Bastian Schweinsteiger miss with their final kick to give Drogba the opportunity to score the penalty to clinch the shoot-out 4-3.
Chelsea goalkeeper Petr Cech also saved brilliantly from Ivica Olic to get his team back into the shoot-out after Juan Mata had missed the Premier League side’s opening effort.
Cech had earlier saved a Bayern penalty from Arjen Robben in the first period of extra-time as the English side defied the odds once more.
Drogba’s winning spot-kick was a fairytale ending for the 34-year-old Ivorian striker, who had been sent off in the final minutes of Chelsea’s Champions League final defeat in Moscow four years ago.
“We have been doing this for years and years. There is a great spirit at the club, we never give up until the end,” said Drogba.
“Today was just amazing. Even if they had scored their penalty in extra-time, Juan Mata said we can get another goal. We all believed.”
Full-back Ashley Cole added: “We should have lost the game with the amount of chances they had.”
“After we lost in Napoli [last 16 first leg], we thought if we can beat them at home then we have the luck and we can win the tournament.
“Even when Robben missed the penalty, we still thought we had time to get another goal and with Petr Cech in goal, we knew we could do it.”
The win capped a rollercoaster season for Chelsea, whose campaign has been transformed since Roberto Di Matteo was appointed as interim manager in March, following the sacking of Andre Villas-Boas.
Di Matteo has guided the club to a Champions League and FA Cup double in the space of a fortnight and is now favourite to land the job on a full-time basis.
Chelsea’s victory – the club’s first European crown and the first by a London team – also means they will qualify for next season’s Champions League as tournament winners despite a poor league campaign.
But the loss left Bayern shattered as their dream of lifting their fifth European title before their own fans at their home ground ended in agony.
A match Bayern had dominated went to penalties after a frenetic finish to normal time, with Müller heading Bayern into the lead seven minutes from time only for Drogba to equalise in the 88th minute.
A dramatic start to extra-time then saw Bayern’s former Chelsea winger Robben miss a penalty – saved by Cech – that was given after Drogba had needlessly brought down Franck Ribery in the area.
For much of the match a Bayern victory had looked the most likely outcome, with the Bundesliga giants revelling in the raucous support of their home fans.
Schweinsteiger was the first to set alarms off in the Chelsea back four, having his shot deflected behind by Gary Cahill after only three minutes.
Chelsea coach Di Matteo had sprung a surprise by handing a first Champions League start to youngster Ryan Bertrand in an effort to cut off the supply lines to Robben down the right flank.
However the wily Dutchman found space with ease in the eighth minute, cutting in from the flank and curling a shot over the bar.
The former Chelsea winger produced the best chance of the half on 21 minutes, teasing the ball past Jose Bosingwa to dart into the area. Robben’s shot was low and hard, but Cech stuck out a leg and the ball cannoned to safety.
The closing stages of the half saw Bayern up the tempo and they carved out a string of openings that they could well have scored from.
On 36 minutes Ribery and Diego Contento combined down the left to send Contento racing away. The left-back whipped in an inviting cross but Müller could only slash his first-time volley wide of the post.
The second half followed largely the same pattern as the first, with Bayern having a goal disallowed for offside by Ribery on 54 minutes.
The Germans were thwarted by some dogged Chelsea defending, with Cole saving the Londoners with a series of telling interventions.
It looked as if it was all over for Chelsea when Müller headed in to put Bayern 1-0 up seven minutes from time. But Drogba had other ideas to set up the extra-time finale.
Chelsea looked to have squandered parity in the fourth minute of extra-time, when Drogba needlessly clipped Ribery to concede the softest of penalties from referee Pedro Proenca.
Once again though Chelsea’s refusal to yield came to their rescue, Cech diving low to his left to thwart Robben’s spot-kick with his legs as the tension rose to excruciating levels.