Drogba shot home after 12 minutes but should have had at least two more as Chelsea passed the Italians off the field in the first half.
Juventus, managed by former Chelsea boss Claudio Ranieri, grew in confidence as the game progressed and though they extended a lamentable away record against English clubs that now features 11 defeats and only two wins from 17 visits they will travel home knowing the tie remains very much alive.
"I'm satisfied with the win but overall I'm not satisfied," new Chelsea coach Guus Hiddink told reporters.
"We started very well, we opened them up through the wings and were very dangerous. But after the goal we dropped back too far and got a bit anxious and Juve came into it much more."
Crisp-passing Chelsea were into their stride from the start and Drogba flashed a header over the bar after nine minutes.
Three minutes later he was handed another opportunity when Salomon Kalou slid a pass through a flat Juve defence. Frank Lampard stepped aside to allow Drogba to shoot low past Gianluigi Buffon.
The Ivory Coast striker, seemingly rejuvenated by the arrival of Hiddink, should have had another soon after when he was left unmarked for a Lampard corner but mistimed his header.
Juve had barely been out of their own half in the opening quarter but showed signs of their potential when Petr Cech had to push wide a crisp Alessandro del Piero shot and Amauri headed wide from the resulting corner.
Chelsea, though, were in the ascendancy, with Lampard controlling midfield and spraying passes short and long. Nicolas Anelka, with 21 goals to his name this season, was happy to ply his trade wide on the left, leaving Drogba, who went close with another header, to occupy the front line.
However, Chelsea lost their momentum and the Italians began to enjoy spells of attacking possession and Cech, although never stretched, became the busier goalkeeper.
Only in the closing minutes, when Anelka hammered a 25-yard effort just past the post, did the hosts come close to claiming the important second goal.
"The players are saying it was one of our best away performances ever against an English team, though I would have preferred to have played less well and got a better result," said Ranieri.
The Juventus manager, who said he was touched by the warm welcome he was given by the home fans, said his team struggled to stay with Chelsea in the opening 20 minutes but got into the game once they relaxed and began to express themselves.
Hiddink looked at the game from the other end. "In the first 20 minutes we could have scored more but after that we dropped back and Juve were clever to play in the gaps we left," he said.
"In the second half Juve dominated. They created danger, though no open chances."
