Anthony Stokes' dramatic extra-time winner against Dundee United at Hampden booked Celtic's place in the Scottish Cup final.
Celtic will face Hibernian on May 26 for the cup.
In a thrilling Sunday encounter where both teams swapped the lead, Celtic midfielder Kris Commons and United skipper Jon Daly grabbed doubles with goals from Tannadice winger Gary Mackay-Steven and Victor Wanyama ensuring 30 added minutes.
However, in the 104th minute, Stokes -- on for Mikael Lustig in the second half -- headed in a James Forrest cross to ensure the Hoops will return to the national stadium at the end of the month hoping to complete a domestic double, with their Premier League title all but clinched.
"It's brilliant, I'm delighted," Stokes told Sky Sports 2. "The boys defended very well, they put their bodies on the line towards the end and we got the victory, and that's the main thing."
Remarkably, Hampden witnessed its second 4-3 semifinal in two days following Hibs' injury-time win over First Division Falkirk on Saturday.
It was asking a lot for both sides to live up to that match but that's exactly what they did.
The game was less than two minutes old when Commons picked up the ball from Victor Wanyama and eased past United midfielder Marc Millar before lashing the ball from 25 yards past Radoslaw Cierzniak.
The Parkhead side, while looking nervy at the back, should have doubled its lead in the 20th minute when Emilio Izaguirre stood a cross up at the back post for fellow full back Lustig, but his header from inside the six-yard box hit the post and John Rankin cleared the ball to safety.
Celtic paid for that miss three minutes later after defender Kelvin Wilson failed to cut out a long ball, handing possession to Daly.
The Irishman played Mackay-Steven down the right channel and, with the Hoops defense posted missing, he played an elongated one-two with teenager Ryan Gauld before bending a shot past Celtic keeper Fraser Forster from inside the box.
United's second goal came when Izaguirre clumsily bundled Mackay-Steven to the ground on the right touchline.
When defender Barry Douglas's in-swinging free kick arrived at the near post, Daly dipped his head amid a packed six-yard box to send the ball past Forster.
The Taysiders' joy was short-lived, though, as just two minutes later, Wanyama rose to bullet Izaguirre's cross past Cierzniak, after Commons had set the Honduras international free down the left.
United responded positively again and a good move ended with Rankin's goal-bound drive blocked for a corner by Wilson, a set-piece that came to nothing.
It was end-to-end stuff and both sides were entitled to the round of applause they received at the interval whistle.
The second half began with midfielder Joe Ledley heading a cross from Forrest over the bar, the Hoops winger back for the first time since March after recovering from a back complaint.
There was an inevitability about Celtic's next goal which came in the 59th minute although it was a rather strange finish by Commons.
Izaguirre again raced up the left, taking a Wanyama pass and when his cross came into the middle the Scotland international did not appear to know too much about it as it bounced off his chest and trundled over the line with Cierzniak's rescue attempt futile.
It was all Celtic by that point. In the 63rd minute, with the United defense stretched again, stopper Sean Dillon cleared Commons' drive from 16 yards away from goal with Cierzniak beaten.
Celtic piled forward in numbers and in the 70th minute only good goalkeeping at his near post from the Tannadice keeper prevented Commons snatching his hat trick after he nipped in to reach Gary Hooper's cross from the right.
However, a minute later there was even more drama at the other end when the unmarked Daly looped a header from Mackay-Steven's cross over Forster and in off the post.
The cup final place, once more, was up for grabs.
Lustig and Wanyama departed to be replaced by Stokes and Beram Kayal, with the latter soon booked for a foul on Flood.
United withstood a late siege by Celtic, with Cierzniak saving from Mulgrew, before Forrest's effort was blocked by Dillon.
Just over 10 minutes into strength-sapping injury time saw Daly rattle the post with a header from a Douglas corner, with substitute Michael Gardyne, on for Gauld, failing to convert the rebound, before Cierzniak made another save, this time from Forrest.
Georgios Samaras replaced Commons just before the teams swapped ends again but it was Stokes who made the headlines when he headed Celtic in front from a Forrest cross.
The second part of injury time saw United keeper Cierzniak make yet another save, this time from Samaras, with two minutes remaining but when the final whistle sounded, the Tannadice side was left with only the consolation of playing its part in a thrilling semifinal, while Celtic look forward to the showpiece finale to the season.
"It's been a great season so far but focus is what it's all about and we want to end the season on a high," Stokes said.
Information from Press Association was used in this report.