The Scottish Premiership side slumped to a 2-0 second-leg loss in Luxembourg on Tuesday as manager Pedro Caixinha oversaw a defeat that he had described in the build-up as "unthinkable".
Olivier Thill was the hero for the hosts – who secured their first win in a UEFA tie in 14 attempts – setting up Emmanuel Francoise to bundle in a 66th-minute opener before his cross-cum-shot from a free-kick deceived Wes Foderingham and found the goalkeeper's bottom-right corner.
Rangers fought to find the breakthrough that would have sent them through on away goals, but Josh Windass saw his late header pushed onto the crossbar by substitute keeper Charly Schinker, who replaced the injured Sebastian Flauss in the first half.
There was still time for first-leg goalscorer Kenny Miller to also strike the woodwork as the Gers' European hopes were ended at the first hurdle.
In the only other qualifying tie of the day, Videoton's 3-3 draw at Balzan saw the Hungarians go through 5-3 on aggregate.