A 3-1 loss at Chelsea on Sunday sees seventh-placed United trailing leaders Arsenal, Manchester City and Chelsea by 14, 13 and 12 points respectively, while there is a six-point gap to Liverpool, who occupy the final Champions League qualifying spot.
"We just keep going," Carrick told MUTV.
"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."
Of more immediate concern to United is the overturning of another deficit when Sunderland travel to Old Trafford on Wednesday night for the Capital One Cup semi-final second leg holding a 2-1 first-leg advantage.
United boss David Moyes admits supporters have had little to cheer of late - his side have lost four of their five games in 2014, two of three at Old Trafford - and hopes to make amends against the Black Cats.
"We will try and do everything we can to get through," Moyes told MUTV.
"I want to give the supporters something to cheer about. There has not been an awful lot. I am fully aware of that."
Carrick added: "We want to play again as quickly as possible.
"Wednesday is another big game. It is a huge game for us. "It is an opportunity to get to a final. You can't take that lightly.
"We will get over this, dust ourselves down and be ready again for Wednesday."
Changes are inevitable for United 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 three times capitalised on United's shortcomings at the back.
"I thought how they played (at Chelsea) wasn't bad," Moyes added.
"The players did well but if you are going to defend the way we did we are going to find it tough.
"We can say bad luck for the first one but not the second and third.
"We were terrible in our defending there. It is just not good enough.
"If we are going to give goals away from set pieces like that it is going to be very hard for us."
England midfielder Carrick agreed with his manager and believes United must do better in the opposition penalty area as well.
"We will have to defend better at set pieces," Carrick said.
"From open play I didn't think they looked dangerous; I didn't think they were much of a threat.
"We needed to create a little bit more ourselves to give ourselves a chance of scoring more goals and give ourselves a chance in games."
Carrick called for the support of the Old Trafford faithful on Wednesday night after an "awful" spell.
"The fans have been unbelievable over the last month or so especially," Carrick said.
"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."
The defeat at Chelsea illustrated further the full scale of the task which faces Moyes after he succeeded Sir Alex Ferguson.
Former Chelsea and England boss Glenn Hoddle believes Ferguson would not have allowed things to deteriorate so quickly and pointed to Moyes' failings in last summer's transfer window, when Marouane Fellaini was the only new recruit.
Hoddle told talkSPORT: "Sir Alex would have known he had to sign a midfielder last summer.
"He was the master of rebuilding a team and reshaping a team, so it never had to be totally dismantled.
"He wouldn't have allowed just Fellaini to come in. Looking back, that was where David Moyes needed to go out and strengthen the squad."
Hoddle believes United need injured strike pair Wayne Rooney and Robin van Persie back swiftly to achieve a top-four spot.
"Six points is nothing, but only if Van Persie and Rooney are playing," Hoddle added.
"They (the United board) have got to give David Moyes time to build something. Manchester United have a plan, they gave him a six-year contract."