Carrick urges consistency from 'fantastic' Man Utd after derby win
The Red Devils were inspired by the change in management as goals from Bryan Mbeumo and Patrick Dorgu inflicted a huge blow on City's Premier League title chase.
United also had three goals ruled out for marginal offside calls and were denied by a string of fine saves from City goalkeeper Gianluigi Donnarumma.
"Great start. No getting away from that," said Carrick. "The boys were fantastic."
Just a second win in eight games in all competitions lifted United into fourth in the table ahead of the later kick-offs.
Carrick's men face another stiff test when they travel to leaders Arsenal next week but this season it is often against the weaker teams they have disappointed.
They failed to beat Wolves, Bournemouth, West Ham and Everton at home under Ruben Amorim, who was sacked earlier this month.
"Consistency is the key to any success," said Carrick. "If you could find that, then you're onto a winner.
"That's our challenge -- we've got to find a way to do that. Big one next week obviously, another another tough one, but on the back of today, we've got a good foundation".