An own goal from David Luiz, plus strikes from Robin van Persie and Javier Hernandez, sealed the points for United over the Blues, who ended the encounter with just nine players after both Branislav Ivanovic and Fernando Torres were handed their marching orders.
The Red Devils had not won a Premier League encounter at Stamford Bridge in 10 years. With that mind, England international Carrick revealed that he was pleased to see that run come to an end.
"It's a huge three points," Carrick told MUTV. "Not winning here for a long time against our biggest rivals, we wanted to put that right and we wanted to win for that.
"Plus the fact that if we had lost today we would've been well adrift. I know there's a long way to go, but it wouldn't have been a good position for us to be in. So for those two reasons it was a big result.
"Over the last three or four years we've had some really great performances here and not come away with what we deserved, so today it was sweet to get the three points."
The victory has drawn United to within one point of leaders Chelsea.