Edin Dzeko scored after just 42 seconds to put City ahead and the big striker struck again after the break before Yaya Toure rounded off yet another miserable night for Moyes.
There was no fortune about this victory from Manuel Pellegrini's team, who are now three points off the top of the Barclays Premier League with two games in hand on Chelsea. If anything, it was a minor miracle City did not win by more.
David Silva, Toure, Fernandinho and Samir Nasri overran and outclassed a weak United midfield and Dzeko - a surprise starter ahead of Alvaro Negredo - did the rest up front. United lacked confidence, tempo and cohesion. There was no cutting edge in attack while the back line was exposed for its lack of pace, as was the case against Liverpool.
Moyes made life easy for the visitors too. Antonio Valencia was left on the bench and Adnan Januzaj did not even make the squad, leaving United with no width at all.
Moyes placed his faith in Tom Cleverley as has been common in the big games this year, and once again the England midfielder failed to deliver.
Things could have been worse had Marouane Fellaini received the red card he deserved for an elbow on Pablo Zabaleta or had David de Gea not prevented Dzeko scoring his third with a superb save.
There were plenty of glum faces by the final whistle, and one of them belonged to Moyes. The statistics do not make for good reading. United are now guaranteed to end the season with their lowest points total in Premier League history. They have been defeated in six home games for the first time since 1978 and they have now lost three successive derbies for the first time in 45 years.
All the noises coming out of Old Trafford before the match were positive. Sir Bobby Charlton said Moyes was "absolutely the right man" for the job. Moyes wrote in his programme notes that it would not be long before United came through their "tricky period" and rose again.
He said his team were not as far behind City in terms of quality as the table suggested, but that theory was shot to bits in the opening minute. Toure sliced open the home defence to find Silva, who would have put City ahead had it not been for a decisive intervention from Rafael.
If United thought the trouble had passed then they were wrong. The cleared ball fell to Fernandinho, who found Nasri in acres of space in the box. The Frenchman's shot beat a stationary De Gea and the ball fell to Dzeko, who was left with the simplest of tap-ins. Moyes sat there looking glum while the away fans proclaimed the Scot a "football genius".
For the rest of the opening quarter, United simply could not get hold of the ball. Silva penetrated the defence but fluffed his shot from 12 yards. Dzeko slid the ball across the box, but Rafael was there to intercept again.
Silva snuck in behind the United back four again and squared for Dzeko, who placed his shot towards the bottom right-hand corner, but De Gea stuck out a hand and saved.
United fans, so supportive this season despite their team's failings, vented their anger as City's attack strolled towards the home defence unchallenged, attack after attack.
Joe Hart was twice called into action, but only to save weak shots from Wayne Rooney and Fellaini. Fellaini then blotted his copy book by elbowing Zabaleta. He was lucky to get away with a booking.
United had one more chance before the break, but Rooney's fresh-air swipe from Rafael's simple pass summed up the hosts' first half.
Moyes brought on Shinji Kagawa for Cleverley at the break but it had no effect on the game. Fernandinho headed over from six yards, but Dzeko made no such mistake soon after.
Fellaini blocked Rio Ferdinand's path as he tracked Dzeko and he was left unchallenged to head home. United threatened momentarily through Welbeck, whose backheel was well saved by Hart from point-blank range.
More substitutions came but nothing changed. Moyes sat in his seat dejected as Toure drove the ball beyond De Gea from 16 yards to wrap up the win in the final minute.