The Gers travelled to the Stade de Geneve in buoyant mood after securing a 2-1 first-leg victory at Ibrox last Wednesday, which was followed three days later by a thumping 4-0 win over Livingston in the Scottish Premiership.
However, Michael Beale's side were second best in the first half and deservedly found themselves a goal down inside the opening 22 minutes when Dereck Kutesa curled a brilliant strike from the left side of the penalty area into the far corner.
Rangers came agonisingly close to restoring parity on the night just after the hour mark when Danilo somehow scuffed an effort from just five yards over the crossbar after receiving and inviting delivery from Jose Cifuentes from the right.
A Gers side lacking width and struggling to get a grip of the game were desperate to hear the half-time whistle, and a huge team talk was required from Beale to shift the dynamics of the contest in their favour.
Whatever was said at the interval certainly did the trick, as Rangers equalised just five minutes into the second half courtesy of a header from right-back James Tavernier.
The Gers captain nodded home his 103rd goal for the club at the back post after meeting a curling cross from the left by Borna Barisic, helping Beale's men move ahead on aggregate.
The momentum was then with Rangers following the equaliser and substitute Cyriel Dessers had two opportunities to add a second, with a header in the 72nd minute looping inches over shortly before clipping the inside of the post with a close-range strike.
Despite Servette's late efforts to net an equaliser in the seven minutes of additional time, Rangers held on the advance to the Champions League playoff round, where they will face Dutch giants PSV Eindhoven over two legs later this month.
PSV, who lost to Rangers in the playoff round last season, have comfortably progressed to the same round this term courtesy of a 7-3 aggregate victory over Sturm Graz in the third qualifying round, winning the second leg 3-1 in Austria.