Match ends, Kilmarnock 1, Rangers 3.
Kilmarnock 1-3 Rangers: Steven Gerrard into last eight after Alfredo Morelos treble
Alfredo Morelos scored a hat-trick as Rangers continued their unbeaten start under Steven Gerrard by beating Kilmarnock in the League Cup.
Gerrard's men moved into the last eight by taking their run without defeat to nine games, and are one match from a sixth successive domestic semi-final.
Morelos headed Rangers in front then added a low finish in the first half.
Borna Barisic's own goal gave the hosts a lifeline, but Morelos tapped in from close range to secure the win.
One negative for Rangers manager Gerrard was the loss of Jamie Murphy, the winger taken off on a stretcher after 15 minutes after falling awkwardly on Kilmarnock's artificial surface.
That means he is a major doubt for Thursday's Europa League play-off first leg with FC Ufa.
Morelos takes centre stage again
Rangers rejected a bid from Bordeaux for Morelos last week, with Gerrard insisting an offer would have to "blow Rangers out of the water" and that he wanted the 22-year-old to sign a new deal.
And Morelos was in the limelight again in Ayrshire, having been sent off in the opening game of the season at Pittodrie.
His day started poorly - albeit in controversial fashion - when he scuffed an attempt from a few yards after Ryan Kent's cross found him in the box. The ball skipped towards goal and, although Stephen O'Donnell lashed clear, television pictures suggested it may have already crossed the line.
At his best, the squat striker is difficult to contain, and after testing debutant goalkeeper Daniel Bachmann with a shot, Morelos headed in James Tavernier's deflected cross to give Rangers a deserved lead.
A second was not long coming. Scott Boyd's defending was questionable, making it all too easy for Morelos to turn him and direct a shot across Bachmann and into the far corner.
A flash of indiscipline threatened to spoil his afternoon, Morelos earning himself a yellow card for needlessly hauling down Alan Power as his own team had possession in the final third.
But soon he was making a more positive contribution again, having a toe-poke and a header thwarted by Bachmann, who also directed Scott Arfield's effort off a post.
The Austrian goalkeeper could only keep Rangers at bay for so long, though, and after a clever move Morelos completed his hat-trick with an easy tap-in from Tavernier's low pass.
Foderingham thwarts second-best hosts
Under manager Steve Clarke, Kilmarnock had beat Rangers twice last season, but his side were very much second best at Rugby Park.
They did work goalkeeper Wes Foderingham on occasion, with the Englishman - deputising for Allan McGregor - tipping over a free-kick from Chris Burke and producing a stunning save to somehow deflect another Burke set-piece over the bar.
A Kirk Broadfoot header following a corner bounced off the top of the crossbar, but the game turned just as Kilmarnock appeared to be running out of ideas.
Jordan Jones scampered clear up the left and his cross was flicked by Kris Boyd on to the knee of Rangers defender Barisic, who could only watch as it bounced into the net.
Suddenly, the hosts had a flicker of hope. Gary Dicker had a header tipped away, while Boyd had a good shout for a penalty after going down under Connor Goldson's push in stoppage time, but the opportunities were too sparse to salvage the cup tie.
'Gerrard's man-management impressive' - analysis
Former Rangers forward Steven Thompson on Sportsound
Morelos has his faults, but he is a player just now playing full of confidence. And that confidence has been given to him by his manager. You can't help but be impressed by the impact Gerrard has had. His man-management of these players is very impressive.
Former Rangers director of football Gordon Smith on Sportsound
Rangers have definitely deserved to go through. They've been impressive. This was always going to be a test because Kilmarnock are one of the hardest teams to play against. Today, though, Rangers have done a great job.
'We've gone out to a good team' - reaction
Kilmarnock manager Steve Clarke: "We are disappointed to go out, but we have gone out to a good team.
"I thought we were still in it to the death but I am not going to be too critical of the players. It is about defending better as a team."
Rangers manager Steven Gerrard: "I'm very pleased. I think the attitude an application of my players has been superb again and every challenge that's been put in front of them, they've answered it , and answered it in style.
"I thought today we were excellent and thoroughly deserved to progress."
- 16BoydSubstituted forNdjoliat 81'minutes
- 8DickerBooked at 21mins
- 6PowerBooked at 73minsSubstituted forWilsonat 81'minutes
- 9BoydBooked at 45mins
- 37ArfieldBooked at 77minsSubstituted forMcCrorieat 79'minutes
- 20MorelosBooked at 60minsSubstituted forSadiqat 79'minutes
- 7MurphySubstituted forCandeiasat 17'minutesBooked at 24mins
- Nick Walsh
- Shots on Target
Second Half ends, Kilmarnock 1, Rangers 3.
Hand ball by Umar Sadiq (Rangers).
Foul by Daniel Candeias (Rangers).
Mikael Ndjoli (Kilmarnock) wins a free kick in the defensive half.
Andy Halliday (Rangers) wins a free kick in the attacking half.
Foul by Kris Boyd (Kilmarnock).
Attempt saved. Mikael Ndjoli (Kilmarnock) right footed shot from the left side of the box is saved in the bottom left corner.
Corner, Kilmarnock. Conceded by Nikola Katic.
Nikola Katic (Rangers) wins a free kick on the right wing.
Foul by Eamonn Brophy (Kilmarnock).
Ross McCrorie (Rangers) wins a free kick on the left wing.
Foul by Stephen O'Donnell (Kilmarnock).
Foul by Ovie Ejaria (Rangers).
Chris Burke (Kilmarnock) wins a free kick in the defensive half.
Substitution, Kilmarnock. Mikael Ndjoli replaces Scott Boyd.
Substitution, Kilmarnock. Iain Wilson replaces Alan Power.
Foul by Umar Sadiq (Rangers).
Kirk Broadfoot (Kilmarnock) wins a free kick in the attacking half.
Substitution, Rangers. Ross McCrorie replaces Scott Arfield.
Substitution, Rangers. Umar Sadiq replaces Alfredo Morelos.
Corner, Rangers. Conceded by Daniel Bachmann.
Attempt saved. Ovie Ejaria (Rangers) right footed shot from the centre of the box is saved in the centre of the goal.
Attempt saved. Ryan Kent (Rangers) left footed shot from a difficult angle on the left is saved in the bottom left corner.
Attempt saved. Kris Boyd (Kilmarnock) header from the centre of the box is saved in the bottom right corner.
Scott Arfield (Rangers) is shown the yellow card for a bad foul.
Foul by Scott Arfield (Rangers).
Jordan Jones (Kilmarnock) wins a free kick in the defensive half.
Goal! Kilmarnock 1, Rangers 3. Alfredo Morelos (Rangers) left footed shot from very close range to the bottom left corner. Assisted by James Tavernier.
Alan Power (Kilmarnock) is shown the yellow card for a bad foul.
Scott Arfield (Rangers) wins a free kick in the attacking half.
Foul by Alan Power (Kilmarnock).
Foul by Ovie Ejaria (Rangers).
Eamonn Brophy (Kilmarnock) wins a free kick in the attacking half.
Foul by Ovie Ejaria (Rangers).
(Kilmarnock) wins a free kick in the attacking half.
Corner, Rangers. Conceded by Kirk Broadfoot.
Delay over. They are ready to continue.
Delay in match Connor Goldson (Rangers) because of an injury.
Corner, Kilmarnock. Conceded by Scott Arfield.