Ronaldo became the first player to score in the first five matches of a European Cup/Champions League campaign for an English team as United qualified for the last 16 courtesy of Wednesday (AEDT) Group F win at Villarreal.
In the first match of the post-Ole Gunnar Solskjaer era, with interim boss Carrick on the sidelines following the Norwegian's sacking, Ronaldo broke the deadlock 12 minutes from the end for his 799th career goal before team-mate Jadon Sancho sealed the victory at the death away from home.
Ronaldo's strike against Villarreal was the third time he has scored the match-winning goal in the final 15 minutes of a Champions League match this season – the most a single player has ever done so in a single season in the competition (excluding extra time).
The 36-year-old has also scored six goals for United in this season's Champions League; only Ruud van Nistelrooy in 2004-2005 (eight) has ever scored more for an English club in a single group phase of the competition.
Carrick – who became the first English manager to win his first match in charge of United (excluding second spells) since 1931 – heaped praise on Ronaldo post-game, telling reporters: "There's not much more I can really say about Cristiano that hasn't been said over quite a number of years.
"It's what he does in the big games and the big moments when you need a goal and you need that little bit of something he's there to deliver. And he's got such a kind of cold, calculated mentality that he's so calm when he gets that and he doesn't snatch at the chances. That's what he's been gifted with. I say gifted with, he's worked very hard at being as good as he is, so delighted to obviously have him and not surprised at all that he managed to come up with a goal for us tonight.
"He played mixed positions. We started him off on the left to put Anthony [Martial] through the middle. And just as the flexibility we have as a forward line, the chopping and changing positions at times, but I thought whatever position they ended up, they defended responsibly and disciplined.
"That [helps] us look a much stronger team and helped an awful lot. In the end, you get your rewards. Like I said before, for Jadon, it's the same thing. If you work well as a team and you put the full package together, you end up getting your rewards and we did tonight."
It has been a turbulent time for United, who had only won two of their previous eight games heading into the matchday five fixture.
Sancho – a big-money signing from Borussia Dortmund at the start of the season – scored his first ever United goal in what was his 15th appearance and from his 11th attempted shot in all competitions.
Carrick added: "I think he'll be better off tonight after playing how he played tonight. The goal for everyone is the obvious one, I think. And for Jadon himself, it'll give him a massive boost.
"It was a big goal for him. But I thought his performance overall, I know how much Jadon likes the ball at his feet and he wants it where he's happiest. You see the smile on his face when he's got the ball on his feet and he's trying to make things happen. But actually, the most pleasant thing for me is the effort and the application that he put on and off the ball.
"And at times he wasn't seeing a lot of the ball, but he was doing a job and he was playing a big part in the team's performance. In the end, that's what happens. You get your rewards and you kind of get what you deserve and you earn it. So I was delighted for Jadon, kind of epitomised the team's performance, really of how much we had to dig in. Grit and determination at certain times.
"But actually how much we enjoyed the ball as well at other times. So I was delighted for Jadon. It's obviously up to Jadon out of what happens next and how he kicks on because he's got all the talent in the world and he'll have all our support, that's for sure."