Things started well for the home side though and they went ahead in the 11th-minute in hugely controversial circumstances.
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Cesc Fabregas seemed to catch Max Meyer with a late challenge, but when no foul was given, Eden Hazard advanced on the Schalke defence and cleverly backheeled the ball into the path of Fabregas, who arrived from the left flank to place a shot underneath the reach of Ralf Fahrmann.
Chelsea should have put themselves out of sight by the time the half-time whistle blew, but Drogba guided a free header straight at Fahrmann before Fabregas blazed a simple chance high over the bar.
Courtois made a superb save on the stroke of half time to push away a vicious strike by Kevin Prince-Boateng and Julian Draxler came within inches of levelling for the visitors after nimble footwork inside the area, but Chelsea held on until the break.
The home side continued to squander chances at the beginning of the second half, with Hazard and Drogba both diverting shots off target, and their profligacy in front of goal came back to haunt them when Klaas Jan Huntelaar steered a finish into the bottom corner from 16 yards to level the game up at 1-1.
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Remy almost made an immediate impact with a strong drive that was headed off the line by Roman Neustadter before Hazard hooked another chance over the bar, but ultimately Chelsea failed to make their chances count.