Linfield, who got this far by beating Faroe Islands side B36 Torshavn on penalties, went behind after Ivorian forward Patrick Vouho rounded keeper Alan Blayney just after the quarter-hour.
The multi-national Cypriot side doubled their advantage on the half-hour when Rwanda defender Edwin Ouon headed in a corner - he had also seen a fourth-minute goal disallowed.
English-born Ghana forward Chris Dickson – formerly of Charlton Athletic – added a third in the second half, having created the opener.
In other matches on Wednesday, Vitali Rodionov scored two injury-time goals as BATE Borisov of Belarus claimed a thrilling 3-2 home win against Macedonia’s Vardar, Bulgaria’s Ludogorets drew 1-1 at home to Croatia’s Dinamo Zagreb, and Slovenia’s Maribor thrashed Bosnia’s Zeljeznicar 4-1.
Ole Gunnar Solskjaer’s Molde benefited from a second-half blitz as they beat Latvia’s Ventspils 3-0, while Montenegrin side Buducnost lost 2-0 at home to Poland’s Slask Wroclaw.