De Boer's solitary strike proved to be enough
Ayr United 0-1 Rangers
Ronald de Boer sent Rangers through to the quarter finals of the Scottish Cup with a late headed goal but the holders were made to fight all the way at Somerset Park.
The Ibrox side dominated the game but they were thwarted by a combination of slack finishing and good goalkeeping from Ayr's Craig Nelson.
Ayr should have forced a replay with minutes to go but teenage substitute Stewart Kean spurned a glorious opportunity following confusion in the Rangers defence.
In the opening minutes Paul Lovering tripped Mikel Arteta just outside the Ayr box but Neill McCann wasted the opportunity, blasting the free-kick well over.
At the other end, Lorenzo Amoruso was penalised for impeding James Grady and it needed a brave header from Craig Moore to clear the free-kick from the danger zone.
But in the 25th minute Ayr were saved by the crossbar after Arveladze flicked a De Boer header over his marker and let fly.
The shot beat Nelson but cannoned back into play off the crossbar and substitute Stephen Hughes headed the rebound wide.
We were one slip-up from going out of the cup.
Rangers manager Alex McLeish
Then Hughes fired in a shot from outside the area but Nelson was behind it.
Minutes later De Boer laid the ball back for Hughes to have another go and, although his effort from the edge of the area was well struck, Nelson again gathered easily.
The home side made a rare foray into the Rangers half and Grady got the better of Moore but was brought down by the Australian just outside the box.
Scott Chaplain fired in the free-kick and Bob Malcolm had to head away under pressure.
As half-time approached Rangers continued to enjoy the lion's share of the possession without creating any clear-cut chances.
The second-half followed a similar pattern with Ayr penned back into their own half and Rangers probing for an opening.
Nelson pulled off a great save to deny Arveladze after he had been played through down the right by Kevin Muscat, with the Georgian's rising drive acrobatically palmed round a post.
Ayr fired in their first shot of the game but Chaplain's effort was straight at Klos.
Ayr First Division side were let off the hook when Arthur Numan laid the ball back to Hughes deep inside the area.
But the young midfielder's weak effort was blocked by Marc Smyth and Arveladze wasted the rebound.
Grady was a constant menace to Rangers
At the other end the ever-dangerous Grady pounced on a loose ball in the Rangers box but Klos dived bravely at the Ayr striker's feet and Numan cleared.
Mark Campbell came up for the corner and headed just over the bar.
Numan's response was to fire in a fierce drive which forced Nelson to make another excellent save.
Grady did well again down the left to set up Smyth but he fired straight at Klos.
Then just when Ayr's stubborn resistance looked to have forced a replay, De Boer popped up in the 79th minute to put the holders through.
Steven Thompson won possession on the right and his high cross was directed over the helpless Nelson by the Dutch international.
Kean and Grady wasted two excellent equaliser chances in the final minutes.
Kean only had Klos to beat but failed to loft the ball over the advancing keeper and into the empty net.
Seconds later Grady fired another good chance over as Rangers held on unconvincingly to make sure their name was in the hat for the next round.
Ayr: Nelson, Campbell, McManus, Craig, Lyle, Smyth, Chaplain, Murray, Lovering, Grady, Whalen (Kean 73).
Subs Not Used: Dunlop, Nicolson, Latta, Dodds.
Booked: Smyth, Grady.
Rangers: Klos, Muscat, Moore, Amoruso, Numan, Arteta (Hughes 21), Malcolm (Thompson 62), Ferguson, McCann (Caniggia 73), de Boer, Arveladze.
Subs Not Used: McGregor, Bonnissel.
Referee: Alan Freeland.