The Hoops have been almost unstoppable this season with 'keeper Fraser Forster setting a new club record of 13 clean sheets in a row as Neil Lennon's side won 14 consecutive games going into the match.
But the holders' hopes of completing a second successive league and cup double ended as Aberdeen inflicted their first domestic defeat since September.
Anthony Stokes fired Celtic into a ninth minute lead as he showed good control and skill to fire past Jamie Langfield.
However, the Dons arrived in Glasgow as the last side to score against the Hoops back in November and buoyed by their League Cup semi-final win over St Johnstone last weekend.
Adam Rooney diverted an Andrew Considine header on to the bar before Russell Anderson ended Forster's run of 19 hours and 23 minutes without conceding a goal as he volleyed Aberdeen level in the 38th minute.
Celtic then found themselves trailing in a game for the first time since November 2 when Peter Pawlett sent a superb angled shot past Forster in the 49th minute.
The Dons held on to book a place in the last eight as they joined Morton, who knocked Celtic out of the League Cup at the same venue in September, as the only teams to have defeated the Hoops domestically this season.
Celtic manager Lennon couldn't hide his disappointment at his side's poor performance.
"We wanted to defend the cup - it's in our DNA as a club - and to go out so early in both competitions is hugely disappointing," the Hoops boss said.
"I can factor in the League Cup as at the time we were in the Champions League and I made some changes for that game, but there are no excuses for today as that was a poor performance.
"We all have to accept the responsibility and the criticism that will come our way.
"You have to give Aberdeen credit as they played well on the day
Going a goal down they showed a lot of character to get back into the game.
"But I've got to look at my own team and that's as poor as we've played here for a long, long time.
"We missed too many decent chances when we didn't work the goalkeeper and there was a flatness about our play today
I don't where that has come from.
"We can analyse it to death but we gave the players all the preparation needed for today and cup ties are like that
We didn't play anywhere near the level that we can."
Aberdeen, who face Inverness Caledonian Thistle in next month's League Cup final, now have the chance to win both domestic cups.
And Lennon believes it is a realistic prospect.
"Absolutely," the Celtic manager said
"They could win both trophies now and I think coming here and winning will give them a huge shot in the arm."
The Dons will be joined in Sunday's draw for the quarter-finals by Rangers, who thrashed fellow League One side Dunfermline 4-0 on Friday night, St Johnstone, Albion Rovers and Dumbarton while Stranraer and Inverness will need a replay following a 2-2 draw.
Hibernian's 112-year Scottish Cup hoodoo continues as Championship side Raith Rovers defeated Terry Butcher's side 3-2 at Easter Road.
Source : AFP
Source: AFP