The Glasgow giants lost all six of their UCL group stage matches last term, but are still in contention to redeem that dismal record this season.
It took only five minutes for Michael Beale’s side to draw first blood in the contest, after David Douline brought down Todd Cantwell inside the box.
James Tavernier smashed in the resulting penalty to further ignite the fervent atmosphere inside Ibrox.
That lead was doubled with only quarter of an hour played, as Cyriel Dessers coolly guided in Borna Barišić's pinpoint cross over the line to score his first Rangers goal.
The hosts were full of confidence and almost increased their lead further when Nicolas Raskin’s speculative long-range strike went just wide of the post.
Servette were chasing shadows in a largely lacklustre first-half display, but were given an unexpected lifeline after Dessers was correctly penalised by the VAR for handball inside the box.
The home crowd did their best to put off Chris Bedia, but the Ivorian smashed home the subsequent spot-kick and halved the deficit before HT.
René Weiler's side had Joël Mall to thank for preserving that scoreline ten minutes after the restart with a fantastic one-handed save to keep out a sweeping strike from Sam Lammers.
From the ecstasy of that save came the agony of a dismissal for the Swiss side, with Douline shown his marching orders.
Cantwell was the victim once again, as the Frenchman was shown a second yellow card for catching his opposite number in an attempt to take a shot.
It was a correct decision by the letter of the law, but it was hard not to have a degree of sympathy for the Frenchman.
Mall kept the visitors in this contest with another superb reflex save to deny Dessers, as Rangers continued to press forward.
Beale’s men failed to increase their advantage, leaving next Tuesday’s second leg in the balance, as Rangers look to reach the UCL group stage in consecutive seasons for the first time since 2010/11.
The Maroons’ expectations are arguably lower, and know that even defeat in this two-legged tie will result in an historic first-ever appearance in the UEFA Europa League group stage.
Flashscore Man of the Match: James Tavernier (Rangers)
