The winner of Sunday's third-round contest, though, was Manchester United — just.
Having stormed into a 3-0 lead and seen City captain Vincent Kompany sent off in the first half, United stumbled after the break against the holders before clinging on to win 3-2 and book a trip to Liverpool.
"We were 3-0 up at halftime and we thought we were through," Ferguson said. "(We) took our foot off the pedal."
City hauled itself back into contention after strikes from Aleksandar Kolarov and Sergio Aguero, whose goal came after the ball was given away by United midfielder Paul Scholes hours after he surprisingly came out of retirement.
"It was a careless performance in the second half and we should have been home and dry," Ferguson said. "We made mistakes and you suffer if you make mistakes."
Similar mistakes have seen the champions lose back-to-back Premier League matches to slip three points behind leader City, and despite the defeat, Sunday's game only strengthened Mancini's belief that he will deliver the club's first English title since 1968.
"If we show the same attitude and strength in other games that we showed against United with only 10 men then yes we can win the league," the City manager said. "If we show the same attitude and strength in other games that we showed against United with only ten men then yes we can win the league. I said we had to improve and I think we are doing so.
"To play 80 minutes against one of the best teams in Europe with ten men and come so close to saving the game was fantastic."
City defender Micah Richards added: "We lost today but in our heads we won!"
But United advanced and were also joined in the fourth round Sunday by Chelsea, Sunderland and Sheffield Wednesday. Chelsea routed Portsmouth 4-0 after scoring all four goals in the second half, Sunderland won 2-0 at Peterborough, and Sheffield Wednesday beat West Ham 1-0.
At the Etihad Stadium, it seemed that United would avenge its 6-1 league loss to City earlier in the season with its own derby rout after a dominant first half.
Wayne Rooney was rampant two days after being forced to deny he was about to leave United due to a rift with Ferguson, scoring the opener after 10 minutes.
The England striker played the ball to Antonio Valencia on the right flank and sprinted forward to meet the return cross, rising above Kompany and Richards to head home.
City's day took another turn for the worse inside two minutes when Kompany was sent off by referee Chris Foy for a two-footed low lunge on Nani. City will appeal against the red card to prevent Kompany serving a four-match ban.
United made its man advantage count on the half-hour, as Patrice Evra crossed from the left into a crowded penalty area, and Welbeck took advantage of Samir Nasri's failed clearance to smash a spectacular volley into the net with his back turned toward the goal.
The third came five minutes before the break after Welbeck ran onto Giggs' ball and was tripped by Kolarov in the area. Rooney's penalty was saved by Pantilimon — deputizing for the rested Joe Hart — but the striker headed the rebound into the net.
United was never so comfortable again.
After Patrice Evra fouled Richards, Kolarov curled his free kick into the net just three minutes into the second half.
The much-anticipated return of Scholes then came, with the midfielder replacing Nani, but United still had a difficult last 30 minutes.
Within minutes — just after United was denied another penalty when Valencia was fouled by Kolarov — Scholes gave away the ball to James Milner on the right. Aguero latched on to Milner's cross and buried the ball into the net after Lindegaard spilled his initial shot.
"We had other chances to score in the end and we didn't concede any chances for them to score," Mancini said.
But United clung on in the face of a City onslaught to secure its first away victory in a Manchester derby since Scholes scored the winner in 2010 in the league.
The 37-year-old Scholes announced on Sunday morning that he was returning to the squad seven months after retiring, with United in the midst of an injury crisis.
"I was surprised, we didn't find out until we got to the dressing room," Rooney said. "It gave everyone a lift. He's a fantastic player."
At Stamford Bridge, Juan Mata put Chelsea ahead against second-tier side Portsmouth at the start of the second half. Ramires scored twice and Frank Lampard completed the victory in stoppage time against the 2008 FA Cup winners.