Trey Nyoni opened the scoring midway through the first half, but Mason Greenwood equalised with a deflected free-kick. Substitute Alexander Isak restored Liverpool's advantage after the interval before Archie Brown brought Fenerbahce level again. Muriqi then headed home the winner in the 79th minute to complete a memorable night for the hosts.
Liverpool manager Andoni Iraola was particularly frustrated by his side's defending from dead-ball situations.
"The main reason I can find to lose the game is we conceded [from] three set-pieces. It's something that we were doing very well in the league, I think we were dealing [with it] quite well until today. We knew it was one of the strengths of Fenerbahce, but we concede from a free-kick, from a corner, even from the long throw at the end."
Iraola felt Liverpool had done enough in possession to win the match, but repeatedly undermined their own work.
"They are goals that have hurt us because we have twice gone in front after having good moments, controlling the game, arriving. We found the goals. But after, we conceded in situations that we definitely should defend better."
He was especially unhappy with the defending for the first and third goals.
"The wall, I don't think we set it properly and the last one Muriqi jumps and wins the ball against three of our defenders, it's something that definitely we have to defend better."
Iraola also rejected the idea that his five changes to the starting line-up were responsible for the defeat, insisting the problem was how his team defended.
"I don't think it was the main reason why we lost, I think some of the players that didn't play on the weekend have played a good game. We have to look more at how we conceded the goals."
For Fenerbahce head coach Dirk Kuyt, the result represented a landmark European night.
"We're very proud of this victory. We've won in this competition for the first time in 18 years, beating one of its most successful clubs. Before kick-off, many people gave us little chance, but we believed in ourselves. We pressed intensely and played with tremendous energy and we're delighted to have given our fans an unforgettable night."
Kuyt also praised the mentality of his players, drawing on lessons from his own time at Liverpool.
"The most important lesson was to never give up, no matter what. That is what I have tried to instil in my players."
The Dutchman added that seeing his side show such confidence against Liverpool had made him particularly happy.
Iraola was left to reflect on a damaging defeat, with Liverpool having twice held the lead but ultimately paying for defensive lapses against a Fenerbahce side whose persistence was rewarded.