Two second-half goals gave Rangers an ultimately deserved win over Kilmarnock to hand the Scottish Cup holders a quarter-final tie with Celtic.
Shota Arveladze slots Rangers' second goal from the penalty spot
Martin Hardie hit the bar early on for Killie and Ronald de Boer headed off the post late in the first half.
De Boer put the visitors in front with a powerful front-post finish from Michael Ball's pass after 62 minutes.
Shota Arveladze secured the win with a penalty shortly afterwards, after he had been brought down by Gordon Greer.
Ronald de Boer had the ball in the net inside the first minute after he had latched on to Mikel Arteta's through ball, but the goal was chopped off for offside.
Rangers had a lucky escape after seven minutes when Garry Hay's ball into the box was met on the volley by Martin Hardie, but with Stefan Klos beaten, the ball came back off the crossbar.
Kilmarnock kept up the pressure on Rangers and forced the visitors into some desperate defending.
At the other end, Shota Arveladze flicked a Michael Ball free kick just wide.
Hardie again came close to opening the scoring after 25 minutes, but he headed narrowly past the post from eight yards out.
Moments later, Arveladze did likewise for Rangers from a Christian Nerlinger corner.
Rangers should have taken the lead on the half-hour mark after Francois Dubourdeau failed to deal with a cross. Nuno Capucho cut the ball back for De Boer who headed onto the post and then blasted the rebound over the bar.
The woodwork denied Rangers again three minutes after the restart when Arveladze rifled in a shot that Dubourdeau managed to touch onto the underside of the bar.
A few minutes later, Klos reacted well to push a powerful Gary McDonald effort wide.
Rangers got the vital opening goal after 62 minutes. Ball beat McDonald on the left before picking out the front-post run of De Boer and the Dutchman finished with some aplomb.
Ronald de Boer celebrates the crucial opening goal
Four minutes later, Rangers were awarded a penalty as Gordon Greer brushed Arveladze aside as the Georgian chased a through ball from De Boer.
Arveladze took the spot kick himself and sent the keeper the wrong way, with the ball going in off the post.
McDonald should have brought Killie back into the game midway through the second half, but he shot tamely into Klos' hands from 10 yards out.
With 10 minutes left, Kris Boyd also passed up an opportunity to reduce the deficit, with Klos gathering gratefully as the ball simply bounced off Boyd.
Eric Skora brought out a fantastic save from Klos moments later, the German having to back-pedal to tip the Frenchman's attempted chip over the top.
Kilmarnock: Dubourdeau, Fowler, Greer, McLaughlin, Hay,
McDonald, Locke, Hardie, Invincibile, McSwegan, Skora.
Meldrum, Boyd, Murray, Dodds, Nish.
Rangers: Klos, Ross, Moore, Berg, Ball, Ricksen, Arteta,
Nerlinger, Capucho, Ronald de Boer, Arveladze.
Ostenstad, Vanoli, Namouchi, Khizanishvili.
Referee: D McDonald