United fell 14 points off the Premier League pace with defeat at Chelsea on Sunday, their fourth loss from five games in 2014.
Moyes' men are six points adrift of the Champions League qualification places, but overturning another deficit is of more immediate concern when Sunderland visit Old Trafford with a 2-1 first-leg advantage.
"The fans have been unbelievable over the last month or so especially," Carrick told MUTV.
"Times have been difficult. Performances have not been that bad actually but results have been pretty awful for us to take.
"We are giving everything at the minute, it is just not going our way. If we all stick together I am sure it will turn soon.
"We want to play again as quickly as possible. Wednesday is another big game.
"It is an opportunity to get to a final. You can't take that lightly."
Moyes added on MUTV: "We will try and do everything we can to get through.
"I want to give the supporters something to cheer about. There has not been an awful lot. I am fully aware of that."
Even if United do overcome Sunderland to reach Wembley, the rampaging Manchester City would be their likely opponents after their 6-0 first leg defeat of West Ham.
It was City's 2012 Premier League triumph on goal difference which inspired Sir Alex Ferguson and his players to regain the title in 2013.
It is clear the squad Moyes inherited needs a radical overhaul, but despite seventh-placed United trailing leaders Arsenal, Manchester City and Chelsea by 14, 13 and 12 points, respectively, Carrick is refusing to throw in the towel.
"We just keep going," Carrick added.
"We don't want to write anything off at this stage. It is not a position we are used to or accustomed to.
"We never give up. We will keep fighting. That is what we do.
"We still believe. We have to try and win as many games as we can and see where it takes us."
Changes are inevitable against Sunderland after Nemanja Vidic earned a three-match ban for his late dismissal for a bad tackle on Eden Hazard and Patrice Evra limped off in the second half.
Moyes might have been contemplating a defensive overhaul anyway after Samuel Eto'o capitalised on United's shortcomings at the back three times .
"If we are going to give goals away from set pieces like that it is going to be very hard for us," Moyes said.