They earned their place in today’s quarter-final draw with a display which, like the curate’s egg, was good in parts, if not entirely convincing.
Of course, in cup competitions, the result is always more important than the performance and there is no question that Rangers did not deserve to progress.
It will be easier to judge their evolution under manager Mark Warburton when they are eventually drawn against Premiership opposition in one of the two main knockout competitions but, for now, their supporters appear to be content that the lesser lights are being put out.
Rangers made five changes from the team which beat Alloa 5-1 at Recreation Park on Sunday, with Tottenham loanees Dominic Ball and Nathan Oduwa making their first starts.
Tom Walsh, Dean Shiels and Nicky Clark also came in, with Rob Kiernan, Andy Halliday, Nicky Law, Kenny Miller and Martyn Waghorn making way for them.
Oduwa had angered Alloa with his showboating but he endured a torrid introduction here, being fouled within ten seconds of the kick-off.
The winger then fell on his backside after miscontrolling the ball on the greasy surface and was then chopped down by Alan Trouten, who was cautioned for his trouble.
Predictably enough, the visitors made all the early running and several half-chances had come and gone by the time the deadlock was broken after quarter of an hour.
Nicky Clark beat Greg Fleming from 12 yards out after Lee Wallace and Jason Holt had sliced open the Ayr United defence.
Three minutes later, Paddy Boyle headed a Clark shot off the line, but the favourites did not have it all their own way.
A hopeful punt upfield by Nicky Devlin should have been dealt with comfortably by Wes Foderingham but, under pressure from Craig Moore, he made a hash of his attempted clearance.
Unfortunately for the striker, on loan from Motherwell, he was forced to shoot from an acute angle and his effort found only the side netting.
Normal service was soon resumed and full-back James Tavernier saw a 20-yarder saved low to his right by Fleming.
A sloppy clearance by Gerry McLauchlan was seized upon by Clark but Fleming parried his shot and Peter Murphy blocked Tom Walsh’s attempt from the rebound.
Rangers’ progress was confirmed beyond doubt two minutes from the interval and, once again, it was a self-inflicted wound.
Brian Gilmour needlessly conceded possession 35 yards from his own goal, Walsh played in Barrie McKay and he beat Devlin before driving home from the far corner of the six-yard box.
There were 47 minutes of regulation time remaining but, even given the long odds then available on an upset, Ray Winstone would not have had a bang on that.
United manager Ian McCall threw on Alan Forrest, the younger brother of Celtic and Scotland winger James, for the second half but he was harshly cautioned for simulation after only 35 seconds following a surging run into the penalty area.
The hosts were nonetheless galvanised by Forrest’s foray and a slack square pass by Dean Shiels almost provided them with a toe-hold in the tie.
It was intercepted by Moore, but a superb challenge by Ball nicked the ball away from him as he was about to pull the trigger.
Rangers responded by moving through the gears and McKay stung Fleming’s fingers with a rasping shot from the edge of the penalty area.
Oduwa will have few fond memories of Somerset Park and he can consider himself unfortunate not to have been awarded a penalty kick when he was taken out by McLauchlan.
Club captain Lee Wallace was booked for a professional foul on Devlin as the right-back burst past him and his discomfort summed up his side’s second-half display.
It may be that they simply opted to see the game out but they played only in fits and starts.
Even so, Clark diverted a driven cross from substitute Martyn Waghorn off the line and Paddy Boyle cleared another shot by the striker off the line. Even then, Moore could have made the closing eight minutes more interesting if, after being picked out by Devlin’s cutback following another lung-bursting run, he could have found the target instead of blazing over from eight yards.
As it was, the Ayrshire club had to settle for a sizeable cheque from what will surely be their highest attendance of the campaign and, like Rangers, they have other fish to fry.