United have made a mixed start under manager Jose Mourinho and have drawn criticism for their lack of killer-instinct in front of goal, having scored 16 times in 11 league games.
"We want to be at the top challenging. We've had some ups and downs along the way, it's not to say that we've given up on winning things just yet," Carrick told Sky Sports.
"It's too early for that and we've got too much quality and too much of a good team and squad to give up on anything.
"We just don't feel we've got the results our performances have warranted; we know it can change... there's still so much to play for and we still believe that when it comes near the end and when it counts we'll be right up there."
Next up for United is a clash against fourth-placed Arsenal and Carrick hoped his team would make a statement of intent against the Gunners at Old Trafford on Saturday.
"Arsenal have got some really good players. When it clicks for them they are a very dangerous team and they've shown in their start to the season how dangerous they can be," he added.
"But I think we can be quite dangerous as well and we probably haven't quite clicked into top gear which, in some ways is a good thing, because we know there's plenty to come."