Drogba sent goalkeeper Manuel Neuer the wrong way on the final kick of the shootout in front of massed Bayern fans in their home Allianz Arena. Bayern's Bastian Schweinsteiger missed the previous penalty, hitting the goalpost.
"It was written, I think, a long time ago," Drogba said to British broadcaster ITV of Chelsea's turnaround since its turmoil in March. "This team is amazing. They never give up until the end."
The shootout was needed after Chelsea goalkeeper Petr Cech saved Arjen Robben's spot-kick early in extra time.
It was awarded for a foul by Drogba, who earned Chelsea an unlikely reprieve.
After 83 minutes of Bayern domination, Thomas Mueller broke dogged Chelsea resistance with a header past the outstanding Cech.
Chelsea's first Champions League title came four years after losing in a shootout to Manchester United.
Drogba succeeded where his captain John Terry, who was suspended for Saturday's finale, failed in missing the fifth penalty in Moscow four years ago which would have given the club's Russian owner Roman Abramavich the Champions League title he has craved.
Abramovich must now make a fascinating decision on the future of interim coach Roberto di Matteo, who took over from the fired Andre Villas-Boas after a last-16, first-leg defeat to Napoli, and inspired a team which then appeared sulky and fading.
Victory also sealed Chelsea's last remaining route into next season's competition which is crucial to its elite status and finances.
Everything seemed stacked against Chelsea when Bayern won the toss to send the shootout to the home, south end of its stadium.
After Bayern captain Philipp Lahm scored first, Juan Mata has his kick saved by Neuer.
Cech then saved Ivica Olic's fourth penalty for Bayern to put the otherwise excellent Schweinsteiger in the spotlight.
He struck the post to Cech's left and covered his face with his shirt.
Drogba stepped up and sealed victory and awaited the adulation of his onrushing teammates.
"He's a hero. Without him we're not here," said Lampard, who scored with Chelsea's third penalty. "I'd love him to stay. What he did tonight he's been doing all his career."
With seven starters from the two teams suspended, Bayern settled quickly against a visiting team set up to absorb pressure.
The Germans' tempo was often dictated by Schweinsteiger, who excelled after collecting a needless yellow card in the second minute for handball.
Toni Kroos, Mario Gomez and Robben all failed to find the target, and anxiety rose in the Chelsea defense when Jose Bosingwa's miscue conceded a corner.
Robben, the former Chelsea winger, threatened again in the 21st when he wriggled through a tiny gap to create a left-footed shooting chance. So often Chelsea's Champions League savior, Cech blocked with his right leg and deflected the ball high up against his right post.
Drogba was being kept quiet before he linked with Frank Lampard to create Chelsea's first good chance in the 37th. Salomon Kalou was teed up to shoot low but Neuer's save was solid.
Gomez, with 13 goals in the competition this season, then wasted two good chances to draw level with Barcelona's Lionel Messi.
First, his control failed when Ribery's misdirected shot flew to his feet, then the big forward wrong-footed Gary Cahill but shot high and wide.
Bayern coach Jupp Heynckes stomped back to the bench throwing his hands in the air.
The second half resumed with the same pattern, and Cole's outstanding defending denied Robben.
The England left-back hunted down Robben's break, then blocked the Dutchman's clear sight of goal from 10 yards (meters) in the 54th. The loose ball found Ribery in an offside position before he thought he'd opened the scoring.
Cole repeated his heroics five minutes later, dashing across to block Robben's shot from 15 yards.
In the 76th, Cech stretched to tip the ball over the bar as Ribery's cross at the goal line squirted up from Cole's lunging challenge.
Cole was then booked for fouling Mueller, who responded by directing a header which Cech gathered.
One minute later, Mueller eluded Cole with a late run and headed home to spark delirious celebrations among the Bayern players and fans.
They underestimated Chelsea's admirable resolve, and Drogba soared to score with a header that Neuer couldn't keep out.
Drogba's taste for the dramatic flared up in the third minute of extra time, as he clipped Ribery's heels in the penalty area.
Referee Pedro Proenca's decision to point to the spot seemed to stun the Bayern fans, and their anxiety was well founded.
Robben fired low to Cech's left, and the Czech 'keeper blocked with his arm and body, then reached across to collect the loose ball.
Bayern came agonizingly close in the 108th, when substitute Olic slipped the ball across the goalmouth but teammate Daniel van Buyten failed to anticipate the simple tap-in.