Second-tier Wigan, who upset City 1-0 in the final at Wembley last May to secure the Cup for the first time, went ahead through Jordi Gomez and James Perch before holding off a rousing City fightback to set up a semi-final against Arsenal.
They were joined in the last four by Sheffield United and Hull City after the third-tier Blades upset Championship (second tier) side Charlton Athletic 2-0 and Hull beat Premier League rivals Sunderland 3-0 in Sunday's other sixth-round ties.
Arsenal had already booked their place in the last four with a 4-1 win over Everton on Saturday.
City secured the English League Cup last Sunday in what fans hoped would be the first leg of an unprecedented quadruple.
Fourth in the Premier League, and needing to overturn a two-goal deficit at Barcelona in the second leg of their Champions League last-16 tie on Wednesday, City were strong favourites to negotiate the visit of Wigan.
But the Championship side, managed by former City striker and terrace hero Uwe Rosler, executed a perfect game plan and capitalised on a tired City display to go 2-0 up after 46 minutes.
"The only chance we had of doing the same here (beating City again)," Rosler told BT Sport, "was to keep the ball. You have to get the ball down ... It is high risk, but our back five, and the midfield, were very good.
"I talked (to the players) about bravery as well. We had Lady Luck on our side, but overall we are delighted."
City defender Martin Demichelis, under the spotlight after recent poor performances, gave away the penalty that allowed Wigan to take the lead in the 27th minute.
The Argentine bought a dummy from Marc-Antoine Fortune at the byline, tripped the French striker in his desperation to make amends and Gomez stroked home the spot-kick.
City were stunned again two minutes after the break, when James McArthur was given time and space to cross and Perch embarrassed a sleepy Gael Clichy at the far post to bundle over the line from two metres.
It took 63 minutes for the home side to seriously threaten the visitors' goal when substitute striker Edin Dzeko headed against a post.
They made their breakthrough four minutes later.
Dzeko used his aerial strength to meet a corner, Micah Richards laid off to Samir Nasri and, from 20 metres the Frenchman drilled a left-foot half-volley through a crowd of players - including Joleon Lescott standing in an offside position - and into the bottom corner of the Wigan net.
"I felt their goal was offside," Rosler said. "Lescott was interfering with (the line of sight of) our goalkeeper."
Seconds later, defender Richards whistled an angled effort inches wide of Scott Carson's upright as City piled on the pressure.
Dzeko came within inches of forcing a replay but Emmerson Boyce produced a remarkable last-ditch block, and then Dzeko again was agonisingly wide with another header.
"It is disappointing," Pellegrini said. "I did not think we could be eliminated here against Wigan, and we had chances to draw, but they played with great intensity and deserved to go through."
Earlier at Bramall Lane, second-half strikes from Ryan Flynn and John Brayford caught second-tier Charlton cold, as Sheffield United became the first third-tier team since 2001 to reach the competition's last four.
Midfielder Flynn put four-times Cup winners United ahead in the 65th minute, when he sneaked in at the far post to nudge home a left-wing cross from Jose Baxter.
One minute, 42 seconds later the home side doubled their lead when defender Brayford's low, 20-metre effort took a cruel deflection off the visitors' Richard Wood and rolled into the net.
At the KC Stadium, Hull reached the semi-finals for the first time since 1930 after striker Sone Aluko missed a first-half penalty against Sunderland.
A game of poor quality was brought to life in the 68th minute when defender Curtis Davies headed the opener.
Four minutes later David Meyler was too quick and strong for Lee Cattermole, who allowed the midfielder to burst past him at the halfway line and slip the ball past goalkeeper Oscar Ustari for the second.
Meyler celebrated by running towards the sideline and headbutting the corner flag in an apparent reference to an incident last week when Newcastle United manager Alan Pardew was sent off for aiming a headbutt at the Irishman during a Premier League game.
"It was just a bit of banter," Meyler told ITV Sport. "The lads were cheering me on ... that's why I did it."
Five minutes later Cattermole was again the Sunderland culprit, inexplicably aiming a back pass towards his keeper but playing the ball straight to Matty Fryatt, who produced a composed finish for number three.
It completed a disappointing week for Sunderland, who lost the League Cup final to Manchester City at Wembley last Sunday, but manager Gus Poyet will turn his full attention now to steering the club away from the Premier League relegation zone.
The FA Cup semi-finals will be played at Wembley on April 12 and 13.