Chelsea had set out with an unfamiliar 4-4-2 formation, using two central midfielders - Michael Ballack and Essien - on the left and right of midfield with the aim of containing what was bound to be an aggressive Juve side.
They were helped in their quest by the exit of playmaker Pavel Nedved after just 13 minutes, the Czech having suffered a rib injury in the early exchanges of what became a tough midfield battle.
Yet it was two touches of class from striker David Trezeguet, who has missed most of the season with a knee injury, which gave Juventus an early lead.
Vincenzo Iaquinta, chosen ahead of Amauri, played a tight pass to Trezeguet on the edge of the box, which the Frenchman controlled on his chest before flicking a delicious return pass over the defence.
Italian international Iaquinta applied the finish that Trezeguet's pass deserved, sliding the ball past Petr Cech to bring Juve level with Chelsea on aggregate.
For a short spell the home side looked likely to add to their lead but the game soon fell flat, enlivened only as half time approached.
Chelsea first came as close to scoring as is possible without the reward, when Drogba rifled a free-kick against the inside of the post in the 44th minute.
The drama didn't end there as the ball span towards the goal and, perhaps, was over the line when Gianluigi Buffon clung on and clawed the ball to safety.
Last year's Champions League finalists were incensed, but they dealt with it in the best possible way, equalising just moments later.
Frank Lampard hit a deflected shot from 35 yards which looped over Buffon and onto the underside of the bar, allowing Essien to tap the ball into the empty net.
It was the security blanket the English side needed and they played like the side with an, in effect, two-goal lead in the second half.
Their efforts were helped by the dismissal of Giorgio Chiellini for two bookable offences, his second coming after 70 minutes by which time the Blues were cruising.
Alessandro Del Piero's 74th minute penalty, awarded after Juliano Belletti used his hands to block a free-kick, gave the game some late life but even then the Blues were barely flustered.
Drogba tapped in Belletti's low cross with seven minutes remaining to secure a good away draw, sending Chelsea through to face, potentially, Premier League foes Liverpool in the quarter-finals.