Amad started on the left flank alongside Matheus Mbeumo and Benjamin Sesko, but struggled badly in an unfamiliar role and was replaced by Noussair Mazraoui at the break as Carrick switched to a back three to protect United's two-goal lead.
Casemiro had opened the scoring with a header from a Bruno Fernandes corner in the 11th minute, and Sesko doubled it before half-time from a Fernandes assist — the Portuguese captain's 19th of the season.
Amad missed two first-half chances, including a free header that should have been converted. He has scored twice in the Premier League this season and has not registered an assist since the draw at Tottenham Hotspur in November.
Despite the scrutiny, Carrick was emphatic in his support.
"He's playing well. There's an awful lot more to the game, playing in a winning team like he has, than worrying about numbers. He gives the team so much."
He was equally firm on the winger's qualities as a player.
"He's got so much talent, so much ability — but his work ethic and his attitude is fantastic as well. He has played different roles at times. I'm a big fan of Amad."
On the half-time change, Carrick made clear there was no personal element to the decision.
"He was unfortunate that we had to change things tactically the other night and that was purely why he came off. He's given the team an awful lot and it's the underbelly of what goes into the performance. It's not always what grabs the headlines that actually provides success. That's Amad's part of it."
Amad faces a challenge to start Sunday's meeting with Liverpool, with Matheus Cunha expected to return from the hip flexor injury that kept him out against Brentford. Gary Neville has suggested Carrick will go with Cunha alongside Mbeumo, Fernandes and Sesko — leaving Amad as a bench option for the club's biggest game of the season.
The result moved Manchester United to within two points of confirming a top-four finish, with three games remaining.