Rashford, who scored United's goal, admitted that his first experience of captaining United was "bittersweet" as Man City won the Manchester derby 3-1.
He tweeted: "Bittersweet to lead out my boyhood team for the first time but lose the game.
"Still in the tie and we'll give it all we've got at The Etihad. Thanks for the support."
Rashford's goal was his 20th of the season in all competitions for club and country and his fourth in three Carabao Cup appearances this season after scoring in previous rounds against Colchester and Chelsea. It was also United's first goal of 2020 — and their only one so far.