Louis van Gaal's men produced one of their finer displays of the season in a 3-0 win over Tottenham at Old Trafford as they maintained hold of fourth spot in the Premier League.
United visit fifth-placed Liverpool for a crucial clash at Anfield, with both teams suddenly eyeing second spot following Manchester City's shock loss to Burnley.
Carrick said the belief and confidence had grown in the United squad following what he thought was their best performance of the season in the win over Tottenham.
"The players are behind him, of course," Carrick said.
"It is just that results prove everything don't they? We can improve on the training pitch, we can get better there, but it is all about what we do on game day really.
"It was probably a big step for us in terms of the size of the game and to play like we did I think will give everyone a big boost and big shot of confidence.
"It shows we are moving in the right direction, which is good, but it is no good settling for that. We still need to keep on that upward curve."
City's surprise loss means Manuel Pellegrini's men are just a point ahead of Arsenal and two above United.
Carrick said second spot had to be the aim for United after City's slip-up.
"Yes, we've got to aim for that," he said.
"We're within touching distance now. You look at the table now and it's so tight, but I suppose really we've just got to look to next Sunday at Liverpool.
"There are going to be ups and downs because that's how it is, especially in the last nine or 10 games.
"Sunday is a big game for a lot of reasons and that is probably all we can focus on at the moment."
