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The hosts led 2-0 thanks to new skipper Jason McGuinness's 18th-minute header and Jay Donnelly's 50th-minute first-time strike into the bottom corner.
Goals from Ivan Trickovski and Elias Charalambous on 59 and 64 minutes brought the Cypriot club level.
Joan Thomas then scored to put his side in control of the tie with the winner.
Cliftonville are the Irish League's last remaining representatives in the competition, with both Linfield and Glenavon having been eliminated at the first qualifying round stage.
The Reds beat Differdange 03 of Luxembourg 3-1 on aggregate in the previous round but were without the suspended Ryan Catney and Martin Murray, who is on holiday, for the visit of Larnaca.
Gerard Lyttle's side face a major uphill struggle in the second leg at the Antonis Papadopoulos Stadium on Thursday, 21 July at 18:00 BST.
Recent signing McGuinness rose highest to power home a towering header from Martin Donnelly's corner as the hosts made a bright start to the game.
Before the break, Jude Winchester and James Knowles were both denied by visiting keeper Rubin Mino as they went close to extending their side's advantage.
Jay Donnelly found the target five minutes after the interval, striking the ball low into the net from inside the area after running onto a ball over the top by David McDaid.
Trickovski, who scored four goals in the last round, met a long ball and beat Mooney to reduce the deficit, then Charalambous beat Jason Mooney with a cool finish.
Thomas got on the end of Trickovski's clever back-heel and fired into the bottom left-hand corner from close range for his team's third away goal.
AEK Larnaca came from two goals down to beat Cliftonville 3-2 in Thursday's first leg of their Europa League second qualifying round tie at Solitude.