The Bundesliga outfit kept their qualification hopes alive with a strong performance at Ibrox as they thoroughly outclassed their opponents to return to Germany with a well-earned victory.
The deadlock in the match was broken in the 16th minute with former Arsenal and Barcelona wideman Alex Hleb orchestrating things down the left.
The Belarus captain played an Arsenal-like one-two with Jeronimo Cacau before squaring the ball across the face of goal.
Despite Pavel Pogrebnyak's failure to bundle the ball home, young midfielder Sebastian Rudy made no mistake at the far post by slotting the ball home for his first career goal.
The visitors thought they had doubled their advantage just two minutes later as Zdravko Kuzmanovic's free-kick cannoned into the back of the net without taking any deflections.
But the effort was chalked off due to the offside presence Matthieu Delpierre, who stood in the way of Gers keeper Allan McGregor.
Walter Smith's side failed to improve after the break and were finished off on the hour.
A quick break from the Germans illustrated the gulf between the sides on the night, as Rudy turned provider to cross for the advancing Kuzmanovic who finished with an authoritative header which gave McGregor no chance.
It could have been worse for the home side but for McGregor, who produced a freak double-save from Kuzmanovic and Pogrebnyak soon after the second goal as Rangers bowed out of European competitions for another season in meek fashion.