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The Reds twice went ahead at Sukru Saracoglu Stadium, via Trey Nyoni and then substitute Alexander Isak.
But goals in response for the home team by Mason Greenwood, Archie Brown and Muriqi secured them the win in Istanbul.
Team
Andoni Iraola made five changes to Liverpool’s starting line-up for the contest.
Joe Gomez and Kostas Tsimikas came into the Reds defence, while there were starts for Victor Munoz, Rio Ngumoha and Dominik Szoboszlai.
Liverpool: Alisson, Gomez (Wirtz, 66), Van Dijk, Jacquet, Szoboszlai, Mac Allister, Tsimikas (Kerkez, 76), Munoz (Isak, 66), Barcola, Nyoni (Koumas, 85), Ngumoha (Gakpo, 66).
Unused subs: Mamardashvili, Woodman, Araujo, Chambers, McConnell, Danns.
First half
Liverpool enjoyed the early chances in the fixture as Munoz threatened on the right flank inside the opening five minutes.
A ball across the box found the winger, though his first dragged shot couldn’t be gathered by Bradley Barcola, while his second was punched clear by Ederson.
Jeremy Jacquet also saw a good opening with a header from a Szoboszlai cross fly the wrong side of the post from the resulting short corner.
Ngumoha was proving a handful for the home defence on the left with several runs, while Munoz again saw another low shot blocked behind as he darted into the box again.
Liverpool had the goal their play deserved in minute 25 as Nyoni struck his first senior professional goal.
A deep ball from Szoboszlai was aimed for Virgil van Dijk, though the ball was defended down into the area and left loose, allowing the midfielder to run onto it and hit a deflected strike into the back of the net.
The hosts, however, responded quickly and instantly hit the bar on their next flurry forward through Irfan Can Kahveci as he slid onto a delivery from the left.
Not to be denied, Fenerbahce were level shortly after.
It was Mason Greenwood who saw his deflected free kick from around 20 yards out wrongfoot Alisson to restore parity with just four minutes between the two goals.
The attacker continued to cause the Reds defence issues as the half reached the latter stages, first dragging a shot wide from a tight angle, before he broke into space to force Alisson into another stern save.
Second half
Liverpool began after the break on the front foot and dominated possession, holding the hosts to openings on the break.
Alexis Mac Allister saw a header saved from a Tsimikas cross, while Barcola’s drive after cutting in from the left was pushed away by Ederson.
Van Dijk also saw a powerful header saved by the goalkeeper from another Tsimikas corner delivery with the Reds pushing to open an advantage.
At the opposite end, Alisson was required to intervene in his first action since the interval, as Brown struck an effort from the left side of the area.
The Reds, though, were the team searching with Munoz producing a clever feint onto his left after a slick move forward to see a shot cleared from underneath the posts.
Fenerbahce responded with the introduction of Muriqi and his first action was to test the reflexes of Alisson with a shot from the edge of the area.
Liverpool made a triple change of their own as Isak, Cody Gakpo and Florian Wirtz entered the tie and two of those combined to earn the lead.
Gakpo broke down the left and stood up a cross for Isak, who jumped highest inside the six-yard box to nod the ball into the net with his first touch and put Liverpool in front.
Similarly to the first half, though, Fenerbahce hit back in instant fashion.
Brown was the scorer as he sprinted in at the back post unmarked to fire home from an N’Golo Kante cross after a short corner.
Minutes later, Fenerbahce had the lead for the first time on the night.
A deep ball into the area following a short throw-in was met by the head of Muriqi and he found the bottom corner to turn the game around with a little over 10 minutes remaining.
Liverpool attempted a response through half-chances for Gakpo and Isak, though were unable to find the key opening and left beaten.