Carlos Bocanegra grabbed his first goal for his new club to cancel out Dalibor Volas' opener, but Rangers could not find a second goal to at least force the tie to extra-time.
They exited the tournament with a 1-1 draw on the night and a 3-2 aggregate defeat, while Celtic suffered a similar fate against Sion and Hearts faced an uphill battle against Tottenham.
Rangers made three changes from their Clydesdale Bank Premier League win over Motherwell for the visit of the Slovenians.
Bocanegra started, after UEFA threw out a complaint from Maribor over his eligibility for the first leg, along with Maurice Edu and Kirk Broadfoot.
Steven Naismith was suspended and youngsters Ross Perry and Jordan McMillan dropped out of the starting line-up following Sunday's 3-0 win.
Rangers looked dangerous early on when Gregg Wylde, who tonight signed a new five-year deal, whipped an inviting cross into the box from the left but Kyle Lafferty just failed to connect in front of goal.
With the ball still in play, Juan Manuel Ortiz went to ground under pressure from Arghus but pleas for a penalty were ignored by the referee.
The home side were threatening again, this time from a Wylde corner into the packed goalmouth, where Nikica Jelavic connected well with the header only to nod wide of target.
Wylde was then the provider yet again, setting up Edu for a header that lacked power and was comfortably gathered by goalkeeper Jasmin Handanovic.
Rangers continued to pile on the pressure and Jelavic released Lafferty through on goal but Handanovic raced off his line to block before Ortiz's rebound effort was deflected wide.
Edu then nodded wide of the post, before Dorin Goian was also off target with a free-kick from 25 yards out.
Rangers were forced to make a change after 38 minutes when Lee Wallace hobbled out of the action and was replaced by Perry, with Bocanegra switching to left-back.
Jelavic then won a free-kick just before the break and connected with Wylde's ball into the box only to nod straight into the grateful hands of the goalkeeper.
Rangers were dealt a major blow after the restart when Maribor opened the scoring on the night after 55 minutes.
Volas collected a long ball and calmly sent a right-foot shot past goalkeeper Allan McGregor from 18 yards.
Ally McCoist made his second change of the night after 67 minutes when Ortiz was withdrawn for the introduction of David Healy as Rangers desperately tried to rescue the tie.
Rangers threw themselves a life-line after 75 minutes when Bocanegra met Goian's knock down from a Wylde corner and rifled home from close range to open his goalscoring account for his new club.
The home side then almost found the back of the net again when Lafferty teed up Healy for the shot and the Northern Irishman saw his point-blank effort blocked by Handanovic.
Maribor suffered a setback when Nejc Potokar was stretchered off as the game entered the final 10 minutes but they held on to book their place in the group stages.