The home side levelled the tie in the 72nd minute of the second leg after Besiktas midfielder Tolgay Arslan curled a finish past Liverpool goalkeeper Simon Mignolet.
Dejan Lovren missed the crucial spot-kick after nine successful attempts, handing Besiktas victory and a place in the Europa League last-16 draw on Friday.
Turkey international Sahan managed to get his team-mates together for a selfie on Ataturk Stadium pitch, as well as posting a cheeky caption.
“Hi Liverpool, this is BESIKTAS!!,” wrote Sahan on Instagram.
Liverpool have little time to recover as the Reds face defending Premier League champions Manchester City at Anfield on Sunday.
Reds striker Sturridge, who wasted a number of opportunities in Istanbul, uploaded a rallying cry to the photo-sharing app following their defeat.
Sturridge posted on Instagram: “We’ll come back stronger. I have no doubts about that. Thanks to all the fans especially the ones that travelled to watch the game live. #redordead”
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