Second-half strikes from Chelsea loanee Patrick Bamford and substitute Kike at the Etihad Stadium on Saturday secured a famous victory for the visitors, who sit second in the Championship.
And Karanka, who saw his side go out of the League Cup against Liverpool on penalties at Anfield earlier in the season, believes his youthful outfit are a match for any team.
"It was amazing, against an amazing squad with an amazing manager, but we were brilliant and when we play like that we can beat anyone," the Spaniard said.
"I'm very proud of the players, the staff, the fans, the whole club. It's impossible to explain my feelings.
"I can't believe we beat them like that, but I am very proud because everyone deserves these things. The crowd deserved that because they are the best."
Over 5,000 Middlesbrough fans made their way to Manchester to witness the upset, which would have been the shock of the fourth round if not for Bradford City's victory at Chelsea.
Despite City missing several chances in the first half – with second-choice goalkeeper Tomas Mejias in fine form for Middlesbrough – it was the visitors who took the lead through Bamford shortly after the interval.
Karanka's team rode their luck, only for substitute Kike to double their advantage in the 93rd minute.
"We have players who can score up front because we have quality and when we are organised we can play against teams like this," added Karanka.
"Their chances in the first half came from our mistakes. We stopped doing that in the second half, had more of the ball, and created more chances.
"I think we showed everyone we were here to win the game after we scored. When the second goal went in we knew we had won."
