It put the Spanish side through on a 2-1 aggregate and, but for some more clinical finishing, they would have won by a wider margin.
Rangers are also likely to face a UEFA probe after their fans clashed with riot police during the match.
Osasuna bossed the game and should have been in front by the interval. Rangers were guilty of some suicidal defending at times but the home side's shooting left a lot to be desired.
The La Liga side could have been in front in the opening minute when Bernardo Romeo raced at the Rangers defence but sent his left-foot shot wide.
Juanlu caused a lot of problems down the left with his pace and Alan Hutton picked up an early booking for a challenge on the winger. He sent one shot across the face of goal but none of his team-mates could take advantage.
Webo had a chance from a Francisco Punal corner but sent his header into the ground and it bounced up kindly for Allan McGregor.
But Juanlu was guilty of a glaring miss when he was left unmarked at a corner and sent his 12-yard shot high over the top.
Nacho Novo had a couple of opportunities for Rangers in the first half but it was mainly one-way traffic.
Two minutes from the break, Charlie Adam should have done much better when Kevin Thomson picked him out at the back post but his header was wayward.
Rangers had a major let-off two minutes into the second half when the ball struck Ian Murray's hand in the penalty area but Danish referee Claus Bo Larsen did not award a spot-kick.
Gavin Rae then missed a chance at the other end when he shot straight at Punal when he had time to do better.
Rangers threw on Chris Burke, Dado Prso and Kris Boyd to try and get the goal they needed but never really looked like scoring.
It was Webo who got the goal for Osasuna after 71 minutes following a mistake by David Weir and the Cameroon player slotted his shot home at the back post. It was enough to settle the tie.