Barca nearly snatched an away goal as early as the eighth minute when Alexis Sanchez's deft lob came back off the crossbar following Lionel Messi's pass.
Chelsea, who had some initial joy in attack, could have bundled in an opener soon after, but John Terry couldn't connect with the ball in a crowded penalty area.
However, the visitors soon began to control possession and they could have gone ahead but for Cesc Fabregas's mistimed rebound following Petr Cech's save from Andres Iniesta.
Despite Barca's dominance, they were often reduced to half chances as demonstrated by Messi's weak header at Cech.
The Spaniards had arguably their best opportunity just minutes before the break, but Fabregas's dink was cleared by Ashley Cole after it had beaten the onrushing Cech.
Out of nowhere, Chelsea took the lead in first-half stoppage time when Drogba rounded off a Chelsea counter attack by slotting in from Ramires's pass.
Barcelona started the second half in the same manner as the first as they made Chelsea work hard for the ball.
Messi continued to torment the home defence, with the Chelsea back line blocking a number of the Argentine's initial shots.
Sanchez was guilty of a glaring miss soon after the restart when he put the ball wide from all of four yards following Fabregas's clever scooped pass.
Barca threatened again as Xavi Hernandez flashed his free kick over, but Chelsea stood firm at the back to frustrate their opponents.
There was still time for late drama as Pedro's controlled shot in injury time came back off the post, with Busquets inexplicably putting the rebound over the bar.