United will face Southampton at Wembley on February 26 after winning the semi-final 3-2 on aggregate, with first-leg goals by Juan Mata and Marouane Fellaini proving decisive.
Tom Huddlestone gave Hull the lead from the penalty spot, only for Paul Pogba to settle the tie with the equaliser, but Oumar Niasse's late goal gave Hull victory on the night.
And Carrick accepted United did not perform well enough despite Jose Mourinho naming a strong starting lineup featuring Pogba and Zlatan Ibrahimovic.
"I wouldn't say we feel relief," Carrick told Sky Sports. "We're disappointed with the way we played and lost the game.
"It is a strange feeling. We wanted to win and keep our run going but we didn't play well at all. We're delighted to be in the final, although it's a strange feeling.
"We didn't believe we were through [after the first leg]. We expected a tough game and for them to come at us and we didn't play well for a number of reasons. We don't like losing and we've lost but we can take the final from it and we're delighted to be there.
"We've got a game on Sunday [in the FA Cup against Wigan Athletic] so we'll get ready for that but we can look forward to the final. It's a big thing as you don't get many chances to win things. We don't like losing and we lost so it's disappointing."
Pogba's goal took him to seven for the season, with only Ibrahimovic more prolific for United, and Carrick felt the France international's equaliser on the night was a key moment.
"It was a big goal for us at that stage and put us in a great position," Carrick said.
"It was nice for Paul to get his goal and hopefully he can go on and score a few goals."
A moody Mourinho, who marked his 54th birthday by booking United's place in the final, appeared to be upset with a number of refereeing decisions that went against his side.
"I just want to say congratulations to my players, because it was a difficult road to be in the final and we are in the final," Mourinho said in a brief Sky Sports interview.
"I don't want to say anything else, it is enough. I am calm, I behaved on the bench, not sent off, and no more words."