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The Gunners showed intent from the get-go, Welbeck sprinting down the left flank early on, but with his teammates unable to keep up, the ex-Manchester United man did well to earn a corner.
Both sides played attacking football and registered several efforts on goal, but the breakthrough duly arrived for the home side in the 22nd minute as Sanchez slid an inch-perfect pass for Welbeck, who was able to slot past Fernando Muslera.
The England international then doubled his and Arsenal's tally on the half-hour mark when he went clean through on goal before beating the Galatasaray goalkeeper with a calm finish.
The match then suffered a delay after fans threw flares onto the pitch, but Arsene Wenger's side weren't perturbed, adding to their goal tally in the 41st minute when Sanchez collected a pass from Mesut Ozil before cutting onto his right foot and drilling the ball into the bottom corner.