Werder Bremen dominated from start to finish, but goalkeeper Allan McGregor kept them at bay as the Gers kept the Scottish flag flying in Europe.
Diego scored the only goal of the game 13 minutes into the second half and Rangers rode their luck a times to gain a famous result.
Rangers were overwhelmed in the first half and it was a major achievement that they managed to get to the interval without losing a goal.
It was embarrassingly one-sided at times as Rangers could not live with Bremen's pace and it was down to the goalkeeping of McGregor that Walter Smith's side survived.
Markus Rosenberg swiped at a shot in a crowded penalty area when he should have done better early on and then Hugo Almeida sent a header over the crossbar from five yards.
Rosenberg brought a fine save from McGregor after 15 minutes when he managed to get a hand to a low shot from the striker.
Diego then split the Rangers' defence with a superb pass, but Tim Borowski's 18-yard volley was a yard too high.
Daniel Jensen then tried his luck after Rangers failed to clear a corner kick, but McGregor was alert to the situation and managed to get a hand to the ball.
Carlos Cuellar was fortunate not to be penalised for a clumsy challenge on Almeida in the penalty area as Rangers were stretched to the limit.
Patrick Owomoyela tested McGregor again with a low shot which the goalkeeper managed to push for a corner.
Rangers' only effort on goal came after half-an-hour when a great pass from Kirk Broadfoot let in Nacho Novo, but his shot was straight at Tim Wiese.
Bremen continued to dominate in the second half and McGregor did well to flip an effort from Jensen over his crossbar with Rangers on the ropes.
Every Bremen player was prepared to have a go, but Rangers knew that the longer they could hold on, the closer they were to the last eight.
But a mistake from Broadfoot gave Bremen the breakthrough after 58 minutes. His header landed at the feet of Diego and he pounced to lash a great left-foot shot past McGregor.
Rangers had a chance to get back on level terms almost immediately, but Steve Davis could not make proper contact in front of goal.
There was a major let-off with six minutes left when McGregor produced the save of the match to touch a point-blank shot from Boubacar Sanogo up onto his crossbar.