Giovanni van Bronckhorst's side was guilty of an underwhelming first-leg performance in Belgium last week, with the Dutchman acknowledging his team was fortunate to only lose 2-0.
But just as it proved crucial en route to the Europa League final last season, the 'Ibrox factor' put them back on course.
Fan expectations will have remained low for much of the first half until a hopeful Borna Barisic cross struck an arm, and James Tavernier converted the resulting penalty with aplomb on the stroke of half-time.
Rangers' tails were up and they appeared rejuvenated in the second period, with Antonio Colak nodding in from close range to level the scores on aggregate.
Extra time looked likely, but another Barisic delivery proved crucial 12 minutes from the end – this time Bayern Munich loanee Malik Tillman beat the stranded goalkeeper to nod into an empty net as the visitors, who had Jean Thierry Lazare sent off late on for two bookings, were vanquished.
Rangers will face either PSV or Monaco – whose second leg went beyond the regulation 90 minutes in the play-off round.