In an attritional encounter on Saturday, both sides had to settle for a point with Victor Valdes and Joel Robles both pulling off superb saves to keep the score goalless.
And although Boro sit just two points above the relegation zone, with a huge clash against fellow strugglers Crystal Palace up next in the league, Karanka - who has previously aired his frustration regarding a lack of atmosphere at the Riverside - was impressed by his side's display.
"More than the result, it was the performance," Karanka told BBC Sport.
"We've played really well against a really good team. We've had the chances and dominated the game, and once again I have to say well done to the players.
"We are confident on the pitch, now we have more options. I think we need to trust ourselves, and the fans have to trust the players. They did today and the fans were brilliant."
While Rudy Gestede could easily have snatched the points late on with a powerful header that Robles did well to parry over, Middlesbrough had to thank Valdes for earlier keeping them in the game.
The former Barcelona man made a string of excellent saves to deny Romelu Lukaku and Ademola Lookman, and Karanka wasted no time in praising his goalkeeper.
"He was really important for us," Karanka added. "It was a pleasure when I convinced him to come and help us at Middlesbrough, because I knew him as a keeper, as a person and I knew his character.
"He's improving a lot. It's not the same to play at Barcelona as it is at Middlesbrough, so he's improving on things he never had to do there."
Meanwhile, Everton boss Ronald Koeman - whose side have now gone eight games unbeaten in the Premier League - believes both teams deserved a share of the spoils.
"It was a fair result. We were better in the first half, but maybe they were more aggressive in the second half," the Dutchman told Everton's official website.
"We dominated at times but didn't do too much, but we did have the best chances to score - one from Romelu and two from Ademola.
"It is a pity. Their goalkeeper was good, but from three chances that good, you normally get the three points."