Steven Craig, son of former Parkhead striker Joe Craig, put the First Division side ahead in the 54th minute with a wonderful solo goal and Hoops fan Martin Scott grabbed a second with two minutes remaining to set up a final meeting against either Dundee United or Raith Rovers.
For the first time in seven years Celtic will finish the season without a trophy and County’s win must have surely ended the chances of Neil Lennon getting the manager’s job on a permanent basis.
But the day belonged to the Highlanders.
The town of Dingwall had emptied with more than its 5000-plus population inside Hampden to cheer on the First Division side.
However, the national stadium, disappointingly, was only around half-full for the lunchtime kick-off.
Lennon had named the same outfield 10 for the third time in succession, with number one keeper Artur Boruc away on compassionate leave resulting in his fellow Pole Zaluska keeping his place in goal.
With Celtic effectively playing 4-2-4, County’s five-man midfield swamped Hoops’ captain Scott Brown and Landry N’Guemo, allowing them good early possession.
The nervy Parkhead side struggled to cope with the Highlanders’ early enthusiasm, however, there was little direct threat to the Hoops’ goal but in the 18th minute Zaluska was glad to tip Scott Morrison’s cross over the bar as the ball curled and dipped towards the top corner.
Celtic survived the corner and Marc-Antoine Fortune had a shot blocked by Iain Vigurs moments later.
Lennon, clearly frustrated by his side’s lethargy, tinkered with his formation, moving Aiden McGeady in to the middle from wide on the left, playing as a support man to Robbie Keane.
Celtic began to impose themselves and Scott Boyd had to clear away a dangerous cross from Fortune who had rolled Morrison in on the right-hand side of the County box.
As the first half continued in stalemate with neither side looking like finding the breakthrough, a situation which frustrated the Celtic fans much more than their Ross County counterparts.
It looked like Lennon, who at times had hurled water bottles to the ground in rage, was trying to prove a point when he substituted the hapless N’Guemo for Marc Crosas just three minutes from the interval.
The break allowed the Irishman to sort his team out but County pressed from the start of the second half, Craig ending one move by volleying wildly over the crossbar from 16 yards.
Any notion that the Victoria Park outfit would pay for those missed chances were blown away when Craig made amends in fine fashion.
He picked up a loose ball 10 yards inside his own half, sped through the middle of the Hoops’ defence past trailing Josh Thompson and Andreas Hinkel and drove past Zaluska.
Moments later, with the Parkhead players still stunned, Scott passed up the chance to make it 2-0 when he failed to control the ball after Hinkel had deflected Michael Gardyne’s cross into his path six yards out.
As Celtic fought back with their fans’ vociferous frustration ringing in their ears, County goalkeeper Michael McGovern made a great point-blank save from Fortune who was then replaced by enigmatic midfielder Paddy McCourt.
Keane somehow managed to miss Brown’s cross from only a few yards before the Dingwall side broke with Thompson having to move quickly to prevent Andy Barrowman getting a clean shot on goal.
The tension mounted as the Parkhead side threw everything forward in their quest for a leveller.
But in the 79th minute Zaluska had to produce a fine save from Richard Brittain’s 30-yard drive as the Staggies looked for what would surely have been the clinching goal.
In the 82nd minute the Highlanders enjoyed a huge slice of luck.
Hoops’ striker Georgios Samaras should have scored when he met McGeady’s cross from the right at the back post but the prostrate County defender Gary Miller deflected the Greek’s close-range shot against the post and the ball bounced to safety.
Moments later, Keane, under pressure from Scott Boyd, headed McGeady’s looping cross past the post.
But more drama arrived in the 88th minute whCounty clinched their final place through the tireless Scott.
The Dingwall midfielder played the ball out to Barrowman on the right and raced in to the box to get on the end of the return pass, bundling the ball over from six yards.
The Celtic fans streamed from the ground and the jeers and boos of those who did remain were drowned out at the final whistle by the incredulous cheers of the Ross County fans who will be back in Glasgow for the final on May 15.