The Blues beat Bayern Munich 4-3 in a penalty shoot-out after the final at the Allianz Arena had finished 1-1 after extra-time. Drogba struck the winning penalty having also headed home an 88th-minute equaliser to cancel out Thomas Muller's opener for Bayern five minutes previously.
"They never give up until the end," Drogba said of his team-mates. "This team is amazing."
The Ivory Coast international added of the victory: "It was written, I think, a long time ago. I want to dedicate this cup to all the managers we've had before, all the players I've played with before."
"When we have this guy in goal you have to believe," Drogba said.
Stand-in manager Roberto Di Matteo, who still does not know whether he will be retained next season, praised the heart of his players who have continually defied the odds.
"We have a group of players with a big heart...that was the only way we could achieve this trophy," the Italian said.
"Drogba has been incredible for this club. It's just been an incredible three months, intense and demanding for everybody."
Asked if Di Matteo should now get the full-time job, captain John Terry said: "Look at that trophy. We hope so. Robbie has been fantastic since he came in.
"It was a superb display from the whole squad. When our backs are against the wall we step up to the mark."
Cech echoed Terry's view of the former assistant coach.
"Whatever happens to him, he has got two fantastic cups. He has done enough to get the job but that is for the board to decide," said the Czech, pointing out Di Matteo also helped Chelsea beat Liverpool in the FA Cup final earlier this month.
Captain for the night Frank Lampard said Di Matteo had helped turn the club's flagging season around.
"It is not a question for this moment. He took us from a struggling team maybe going out of the Champions League and now we have won it, so just look at that," Lampard said.
The Londoners finished sixth in the Premier League, their worst placing for a decade, but the Champions League triumph ensured they would make next season's top European competition.
Media reports have linked Chelsea with former England manager Fabio Capello and neither Di Matteo and owner Roman Abramovich have spoken publicly about the future.
"I'm enjoying this moment. I won't talk about it now," former Chelsea midfielder Di Matteo said.
Chelsea's suspended captain Terry missed the club's penalty shootout victory but still described the win as the best moment of his career.
"This is an incredible feeling," Terry said. "I've waited eight or nine years for this moment and it wipes out all those bad memories from this competition. It's 100 per cent the top moment in my career.
"In the past we've been knocked out in the last minute, lost on penalties to Manchester United in the 2008 final and twice we lost to Liverpool in the semi-finals but this victory takes a big weight off everyone's shoulders..
"I can live with the fact I didn't play because we have won. That's all that matters to me because I care so much about this football club.
"It's probably harder watching. You know then what the fans go through and it's really frustrating," he added.
"But it's great to see us win the trophy I think we have deserved, to see the smile that was on owner Roman Abramovich's face after the game and to see the fans at the end.
"We've won the Champions League and the FA Cup this season so we we'll go down as the best ever team in Chelsea's history and that's incredible."
Terry thanked European ruling body UEFA for allowing him to join the team in the post-match celebrations and letting him lift the biggest trophy in European club football.
"To be involved was incredible," said the former Chelsea youth team product. "If I hadn't been involved it would have been very hard to take.
"Credit to UEFA for making the right decision to allow all four of our suspended players to be involved."
Even when Chelsea looked down and out earlier in the competition Terry said he had faith his team could land the Champions League trophy for the first time.
"You look back at the Napoli game in the first knockout round when we were 3-1 down from the first leg," he said. "Everyone wrote us off but I believed in us and within the squad we all believed in each other. It's great we've got so many big players for big occasions, players like Didier Drogba, Fernando Torres, Frank Lampard and Petr Cech."
Lampard began to think there was no way back for Chelsea as the minutes ticked down with Bayern leading 1-0, but when the match went to a penalty shootout he had no doubts the trophy was coming to London.
"I admit when we were trailing 1-0 with about a minute left, I wasn't so sure," said Lampard.
"Once it went to penalties, I thought there was no way we were going to lose. I thought, we are going to win this."
But Lampard said the heartbreak of losing in the final four years ago to Manchester United on penalties spurred his team to victory.
"It wasn't going to be a repeat of Moscow, we couldn't have stood that again," he added.
"I really wasn't sure there was a way back and I was thinking well, perhaps it wasn't meant to be after all ... then Didier equalised with a minute to go and Petr (Cech) saved Arjen Robben's penalty in extra time and I thought, 'we are going to do this'.
"And when the penalties started I was sure."
The 33-year-old heaped praise on Drogba, whose contract is set to expire and could well have played his last match for the club.
"I have to say, Didier was the main man tonight, and he dug us out of a hole there.
"The man is a Chelsea legend, he is a hero, without him we're not here. He scores the goals in the big games."
Asked if this victory could mark the break-up of a side that have largely been together for most of the last decade, he said: "No, I don't see why it should.
"Teams are always changing, but why should we stop now? We want to carry on, move forwards. We've won the FA Cup and now the Champions League and the determination and spirit we have shown, it’s been fantastic."