United have not lost in the top flight since a disappointing 4-0 defeat to Chelsea in October, but have won just three out of their past nine league matches.
Jose Mourinho’s men left it late to salvage a point on Tuesday as Zlatan Ibrahimovic’s injury-time penalty secured a 1-1 draw against Everton.
That result left United four points adrift of rivals Manchester City, who occupy fourth, but Carrick says Pep Guardiola’s men can be caught.
“You cannot give up on anything and results can go funny ways sometimes,” Carrick told MUTV.
“We are obviously giving ourselves the tougher task but we are not giving up.
“We just cannot find a way to find that first goal and to break teams down and make it easier for us. Obviously, it is something you look at and you try to improve on and it is not just an easy thing you can try and just switch on.
“I am sure we will get there. I am sure we will improve. Generally, the performances have been very good at home.”