Carrick struck in the 3-0 win.
"I think it was very important for us to bounce back with a good win," Carrick told MUTV.
"Everton always had a threat with Louis Saha up there and Yakubu coming on, so we had to keep pushing for another goal. It was nice to see the second and third go in. It meant we could play our football and relax a little bit.
"It's been a tad frustrating when we haven't been able to kill teams off at home. We've been creating enough chances to do it, but the goals just haven't been going in for us.
"Thankfully they did against Everton and although we could have had a few more, I think the scoreline reflects fairly on the game."
United's ninth win in thirteen league games so far means the champions are well-placed to assail Chelsea's lead at the top of the table over the next two months, even after losing the so-called six-pointer at Stamford Bridge.
"Losing to Chelsea was a blow," says Carrick, "but we took a lot of confidence from the game. Today (Saturday) showed that result (0-1) didn't knock us off our stride. We know what we can do and we know we can still be up there where it counts in the league."