Carrick's appearance on Sunday, in otherwise unremarkable final Premier League game of the season, was his 464th and last for United.
The 27-year-old said it was important the club keep silverware coming in as they emerge from a post-Ferguson dip in form.
"We have to be fighting hard to win something and we have that opportunity next week [in the FA Cup final against Chelsea]".
Manchester United manager Jose Mourinho is reportedly looking to axe four stars in order to fund a £150 million (R2.49 billion) spending spree in the next few months. I'm grateful for everything you've done for me, thank you. It played on my mind a little bit, but apart from that it's happened and there's nothing I can do about it.
It was also United assistant manager Rui Faria's final game at Old Trafford after the club's announcement on Saturday that he would step down at the end of the season.
Mourinho said: "Moving forward I will organise my coaching staff in a way where the assistant manager figure doesn't exist".
"That was what I told the players: imagine in this match [against West Ham] we need a point to be champions".
"Lastly it's been a tough week for everyone at the club".
"But it is a hard role, you need somebody to talk to because you can't bounce things off people if they're not there". "If you take your foot off the gas, that's you out".
"I know that Romelu will be back for Tuesday because he's in the last week [of rehabilitation] and we have to see but Anthony I have no more". Huge players, bigger than me, and the club has moved on and still been successful.
He bears no hard feelings towards his old boss and feels he has nothing to prove against the United manager at the weekend.
Fred, who is fighting for a Brazil squad place for the upcoming World Cup in Russian Federation, fits the bill in what Mourinho is looking for in a central midfield player and also has plenty of Champions League experience.
"It was nice to play and win, the reception was very special".