The Ivorian striker lashed the ball into the top corner from the edge of the area in the 47th minute to secure the victory for the hosts.
Barca's last defeat in the Champions League was in March 2005 when they lost also against Chelsea at Stamford Bridge 4-2 in the first knockout stage.
The Premiership champions maintained the top spot of Group A with perfect nine points from three games.
Barca and Werder Bremen, who beat Levski Sofia on the same day, are second and third respectively with four points each with the title holders leading on goal difference.
Chelsea played without their two big keepers Petr Cech and Carlo Cudicini, handing a slot to Portuguese goalie Hilario who put on an outstanding performance.
Hilario blocked two clear-cut chances when he denied two close-range shots from Xavi and Lionel Messi.
Chelsea's closest opportunity was foiled by Barca defender Rafael Marquez who cleared the ball just before Andriy Shevchenko pokes home from one meter.
But Drogba snatched the lead just two minutes after the restart when he controlled Ashley Cole's long pass with his back to the goal, span around his marker Carles Puyol and curled a superb finish past Victor Valdez.
The Londoners had many chances to stretch their lead but poor shooting from Shevchenko and Michael Essien left the score line as it is.
In Germany, Bremen labored to a 2-0 victory over Levski to stay in the race for qualification.
Naldo scored just before the interval through a low free-kick from the edge of the area to cap a dominating display from the former Bundesliga champions.
Brazilian playmaker Diego sealed the victory in the 73rd minute when he raced on to Ivan Klasnic's first-time through pass before rounding the keeper the poking home.