Boro were unable to book a place in the League Cup semi-finals after losing 2-0 to Roberto Martinez's side.
First-half goals from Everton's Gerard Deulofeu and Romelu Lukaku ended Boro's hopes of booking a place in the semi-finals for the first time since 2003/04, where they would go on to win the competition.
Karanka suggested that while they need to move on from the defeat, and switch their attention back to the league, they can look back on their progress with plenty of satisfaction.
"Now we have to forget the cup, to be pleased and proud of the run in this Capital One Cup, playing against Manchester United and against Everton and showing everybody that we are a good side on TV," he said.
"Now we have to be focused on the game on Friday. That is our challenge, the league," he added.
The Riverside Stadium outfit have made an excellent start to the Championship season. Saturday's win over Huddersfield Town has left them second in the table, two points behind league leaders Brighton & Hove Albion.
However, their next four games are likely to indicate whether Boro are serious contenders for promotion to the Premier League.
An away trip to seventh-placed Ipswich Town on Friday is followed by consecutive home games against sides currently sat inside the play-offs places - Birmingham City and Burnley - before they complete what could be a gruelling run with a trip to the Amex Stadium to face Brighton.
Should Karanka's men come through that period unscathed, then they would have not only strengthened their own promotion bid, but also have dented their rivals' hopes of going up.