City defender Josko Gvardiol opened the scoring in a low-quality encounter at the Etihad but Bruno Fernandes levelled from the penalty spot in the 88th minute and Amad Diallo struck two minutes later to seal a 2-1 win.
Defending champions City -- who have dominated English football over the past decade -- now have just one win in 11 games in all competitions and Guardiola is desperately searching for answers.
"I don't have a defence, I'm the boss, I'm the manager. I have to find a solution and I don't find a solution," said the Catalan.
"This is a big club, when you lose eight out of 11, something wrong is happening. What can I say? The schedule is tough, the injured players? No.
"I'm the boss, I'm the manager and I'm not good enough. It's as simple as that."
Guardiola's men trail leaders Liverpool, who have a game in hand, by nine points and sit outside the Champions League places, in fifth spot.
Injury-ravaged City were far from their best at the Etihad but appeared to be on course for a vital win as the clock ticked down against their neighbours but they self-imploded.
Matheus Nunes' sloppy back pass was cut out by Diallo and the Portuguese midfielder then chopped down the United winger in his attempt to make amends.
Fernandes coolly converted from the penalty spot to equalise.
Diallo then raced onto a simple long ball over the top of City's flat-footed defence, finishing neatly to complete a remarkable turnaround.
City midfielder Bernardo Silva was scathing about his team's performance.
"If we make these stupid decisions with three or four minutes to go you deserve to pay for that," he said. "Today in the last minute we played like under-15s."
But the win is a huge relief for new United boss Ruben Amorim, who had won just one of his first four Premier League matches.
"I think for the fans it has a deep meaning, especially in this context, in this moment," the former Sporting Lisbon boss said after his first experience of a Manchester derby.
"We had a difficult moment one week ago against Nottingham (Forest) at home with a storm. It was a really tough moment after the game. Today, we are more happy, but it's three points that we have to continue."
United finished the day 13th in the Premier League table -- where they started -- after Tottenham hammered bottom club Southampton 5-0.
Chelsea charge
Enzo Maresca's high-flying Chelsea beat Brentford 2-1 at Stamford Bridge to close to within two points of long-time leaders Liverpool, who have 36 points.
The home team dominated the first half and got the reward they deserved shortly before the break when Marc Cucurella nodded home Noni Madueke's cross.
Nicolas Jackson doubled Chelsea's lead in the 80th minute and they held out for the win despite a late strike from prolific Brentford forward Bryan Mbeumo.
It is a fifth straight win for Chelsea, who look increasingly like being Liverpool's main challengers this season, with Arsenal and City faltering.
But Maresca is still being coy over his side's title chances.
"It's not about how many games we win," he said. "It's about being realistic. There are things we have to do better. That's why I said for me we are not ready."
Tottenham's romp on the south coast was the latest chapter in a topsy-turvy season for Ange Postecoglou's men, who have now won seven and lost seven of their 16 top-flight matches.
James Maddison opened the scoring in 36 seconds and further goals followed for Son Heung-min, Dejan Kulusevski and Pape Sarr in a staggering 25-minute spell.
England midfielder Maddison grabbed a second goal deep into first-half stoppage time to make it 5-0 at the break but there were no additions in the second half.
In the early kick-off, Ismaila Sarr scored twice as Crystal Palace moved four points clear of the relegation zone following a 3-1 victory at Brighton.
Earlier Sunday, Wolves, in the relegation zone, sacked manager Gary O'Neil after a fourth straight defeat in the Premier League.
