Despite struggling throughout to contain the mercurial talents of former Manchester City striker Georgi Kinkladze, Rangers stunned the Cypriot champions with vital away goals from Nacho Novo and Fernando Ricksen to take control of the qualifying round tie and only Nikos Frousos' strike took some of the gloss off the Gers effort.
The Cypriots made an impressive start but, despite their possession, it was the visitors who forced the first save when, in the 11th minute, Novo's volley was touched round for a corner by Antonis Georgallides.
Rangers continued to press and they should have been ahead in the 15th minute when Novo sent Ian Murray clear in the Famagusta box but, with only Georgallides to beat, the midfielder blasted over.
Frousos twice went close for the home side but scuffed his 19th minute header wide and then, after 36 minutes, beat Ronald Waterreus only to see his shot came back off the post and back into the Dutchman's grateful grasp.
After soaking up more Famagusta pressure, Rangers took a shock lead in the 66th minute. Dado Prso created the opening, and although Georgallides parried Ricksen's overhead kick, Novo pounced on the loose ball to fire Alex McLeish's side in front.
Five minutes later Rangers were two ahead when Prso played an inch-perfect pass to Ricksen who slipped the ball under Georgallides for Rangers' 200th European Cup goal.
But Famagusta gave themselves a lifeline in the 72nd minute when Frousos took advantage of some indecision from Julien Rodriguez to bury a low shot beyond Waterreus.