All three goals came in the first-half as James Tavernier had Rangers ahead early from the penalty spot before Cyriel Dessers doubled the advantage in his competitive home debut.
Servette were awarded a penalty of their own before the break following a VAR check with Chris Bedia converting to reduce the deficit.
Beale named three changes to the team which faced Kilmarnock in the Scottish Premiership on Saturday evening as Ryan Jack, Todd Cantwell and Danilo came into the starting XI with John Lundstram, Kieran Dowell and Abdallah Sima named on the bench.
Rangers started at Ibrox with Butland in goal and a back four of James Tavernier, Connor Goldson, John Souttar and Borna Barisic.
Nico Raskin was joined in midfield by Ryan Jack and Todd Cantwell while summer signing Danilo made his first start for the club in attack alongside Sam Lammers and Cyriel Dessers.
It was an extremely positive start from Gers as they sent the ball forward straight from kick-off. James Tavernier drove down the right before finding Cyriel Dessers who touched it into the path of Todd Cantwell but his powerful volley blocked on the edge of the box.
With just five minutes on the clock, Gers were awarded a penalty as they continued to pile the pressure on the Swiss side early in the match. Cantwell was fouled inside the box as he looked to get on the end of another good move for Rangers.
James Tavernier stepped up to the spot and the captain once again showed his calmness from the penalty mark as he beat Joel Mall to give Michael Beale’s men an early lead.
Rangers couldn’t have asked for a better start in this qualifier as Cyriel Dessers marked his competitive Ibrox debut with a goal as the match approached 15 minutes.
Centre-back John Souttar started the move as he stepped forward driving into the Servette half, before feeding the ball out to Borna Barisic on the left. The Croatian whipped an inviting cross into the box with Dessers on hand at the back post to finish beyond Joel Mall.
Nico Raskin came close shortly after with his strike from outside the box drifting just wide of the post. On the break, Foana Boubacar tried a long range effort for the visitors but it was gathered comfortably by Jack Butland.
Ibrox was in full voice as Rangers continued to threaten in this first-leg tie. James Tavernier played a ball into the area with Dessers managing to touch it into the path of Danilo. The strikers’ shot was on target, but was held by Mall at the second attempt.
With 38 minutes on the clock, Nico Raskin showed his strength to break forward before playing a one two with Cyriel Dessers with Raskin receiving the ball on the return and driving into the Servette box. The midfielder took a touch before sending a ball in search of Sam Lammers but his effort flashed inches wide of the target.
Following a VAR check, Servette were awarded a penalty as referee Donatas Rumsas was called over to view footage after a free-kick for the visitors appeared to come off the hand of Cyriel Dessers inside the box.
Chris Bedia took to the spot for Servette and sent a low left footed effort beyond Jack Butland in the Rangers goal to get a goal back for the Swiss side.
Into the second-half, Joel Mall pulled off a fantastic save to deny Sam Lammers a goal after another fine attacking move from Rangers. Nico Raskin and Cyriel Dessers were involved as the former turned well to find Raskin who spread it out wide to James Tavernier.
Sam Lammers met the cross in the area and hit it first time towards goal, but he was somehow denied by Mall who managed to get a hand to deny the Dutchman a goal.
Servette were reduced to 10 men just before the hour mark as David Douline was shown a second yellow card for a late foul on Todd Cantwell.
Beale turned to his bench for the first time of the evening as he introduced Abdallah Sima for the final 25 minutes at Ibrox as he replaced Danilo.
Mall pulled off another strong save to prevent Rangers from increasing their advantage as James Tavernier found Dessers from a corner but his first time strike was matched by a strong hand from the goalkeeper before Barisic’s follow-up effort was blocked.
There was a Rangers debut for Jose Cifuentes as he took to the field in the 76th minute alongside Kieran Dowell, with Ryan Jack and Sam Lammers making way.
Sima had a shot held by Mall in the final 10 minutes of the match as Beale’s men looked to increase their advantage before the manager turned to the bench again with minutes remaining as Scott Wright and Ianis Hagi came on in place of Cyriel Dessers and Nico Raskin.
With six minutes added at Ibrox, Rangers continued to search for a third goal but were unable to add to their advantage and head for Geneva next Tuesday with a 2-1 lead in this qualifying tie.
RANGERS: Butland, Tavernier, Goldson, Souttar, Barisic, Jack (Cifuentes, 77), Raskin (S. Wright, 87), Cantwell, Lammers (Dowell, 77), Dessers (Hagi, 87), Danilo (Sima, 65)
Subs not used: McCrorie, K. Wright, Lundstram, Sterling, Balogun, King, Devine.
Servette: Mall, Rouiller (Baron, 90), Cognat, Boubacar (Diba, 64), Kutesa, Mazikou, Severin, Guillemenot, Douline, Bedia (Rodelin, 79), Vouilloz.
Subs not used: Marwan, Besson, Pflucke, Kaloga, Henchoz, Ouattara, Touati, Onguene.