Ronaldo broke the deadlock in the 78th minute with a clever finish after an error from Emmanuel Capoue, before Jadon Sancho rifled in his first goal for United in the 90th minute.
The result means United progress in Group F, sitting top of the pool with 10 points from five games, ahead of second-placed Villarreal with seven points and Atalanta in third with six points after their 3-3 draw with Young Boys.
Carrick, who stepped in following Ole Gunnar Solskjaer’s sacking on Sunday, said: “That one meant a lot. It was a tough, tough game, they’re a good team. They really test you and make you work.
“Its a terrific night for the players, a terrific night for the club, to go through in the Champions League is amazing. That ones for Ole that.”
He added: “When Ronaldo gets that chance, you know he’s going to put it away, it just ticks another one for him on his list.”
Both sides are through in their group with Malmo and Zenit St Petersburg drawing 1-1.
Barcelona missed a chance to clinch their last 16 spot and almost let it slip after a 0-0 home draw with Benfica, who had Haris Seferovic miss a golden opportunity late. Already-qualified Bayern Munich won 2-1 at Dynamo Kiev.
Lille moved top in a tight Group G, with a 1-0 win over Salzburg, while Sevilla defeated Wolfsburg 2-0. All four sides can still progress.