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 should have gone ahead but for Cesc Fabregas's mistimed rebound after Petr Cech had saved 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.
The home side's goal came in first-half stoppage time when Drogba rounded off a Chelsea counter attack by slotting in from Ramires's pass.