However, the Blues rode their luck in the second half as the Serie A side dominated possession and created several clear openings.
The game marked Juve boss Claudio Ranieri's return to Stamford Bridge while Guus Hiddink led out his Chelsea side for the first time on home soil.
The Blues seem to have been revitalised since the Dutchman's arrival, securing a vital 1-0 win at Aston Villa on Saturday and producing another fine display, particularly in the opening 45 minutes, against Juve.
Chelsea started brightly and created the first opening in the fifth minute when Salomon Kalou's curling effort forced Gianluigi Buffon to make a fine save.
Three minutes later Kalou found Jose Bosingwa who whipped in a cross from close to the corner flag. Drogba launched himself at the ball but could only head it over the bar.
However, Chelsea took the lead in the 12th minute. Kalou's clever, precise pass found Drogba who beat Frank Lampard to the ball before firing it past Buffon.
Chelsea threatened to run riot as Juve buckled and Olof Mellberg was forced to block Nicolas Anelka's shot before Drogba headed Lampard's corner wide.
Midway through the first half Juve finally upped their game and they almost equalised with former Blues star Tiago teeing up Alessandro Del Piero who forced Petr Cech to make a smart save.
Chelsea were all at sea from the resulting corner but Cristian Molinaro was just unable to get a good connection to the ball as it whistled across the six-yard area.
If Chelsea had edged the first half, Ranieri's side dominated the second period and were unfortunate not to grab an equaliser.
Marco Marchionni fired in an ambitious shot just past the hour mark that was only just too high and Del Piero flicked a header wide before Cech was forced to keep Amauri's header out.
Both sides almost scored in the closing minutes with Anelka whistling a shot millimetres wide before substitute David Trezeguet failed to find the target with his first touch after coming on.
With the last kick of the match Pavel Nedved's shot was deflected narrowly wide but the tie is still evenly poised going into the second leg in Turin in a fortnight's time.