Connor Goldson's close-range finish in the 41st minute put Steven Gerrard's side within touch of group stage qualification, but they may feel as though the two-legged affair should already be over.
The match-winner himself headed over a good opportunity prior to his goal while Alfredo Morelos was repeatedly frustrated in his attempts to increase the margin.
Rangers will, however, be confident of protecting their lead after encountering few threats at Ibrox.
It was dominant from the outset and should have been ahead when James Tavernier crossed for Morelos in the 11th minute, but the striker was caught in two minds and mustered only a tame connection.
The 22-year-old Colombian remained a lively presence despite the miss and it was his downward header from Ryan Kent's corner that Aleksandr Belenov failed to hold shortly before half-time, enabling Goldson to poke home at point-blank range.
Allan McGregor's only real moment of concern at the other end came immediately after the goal as he was forced to parry a Dmitry Zhivoglyadov drive from inside the area.
Two more opportunities for Morelos came and went before Kyle Lafferty, signed from Hearts on Wednesday, was called upon to make his second debut for the club he left in 2012.
Appeals for a penalty following an apparent trip on substitute Glenn Middleton later went unrewarded but, still unbeaten during his short stint in charge, Gerrard will expect to end Rangers' seven-year wait for European qualification in next Friday's (AEST) return leg.