Neither keeper was tested in the first half at the Riverside, but Grant Leadbitter's free kick around the hour mark opened the scoring before Alvaro Negredo and Marten de Roon made the game safe.
Boro spent the final 45 minutes with ten men after Dani Ayala was sent off soon after Leadbitter's opener, but Karanka admitted that he does not know if he will appeal the decision or not.
'The FA Cup is really important'
"I think the second half was completely different," the Spaniard noted after the game. "I said at half time that it was like we were playing a friendly game and the FA Cup, for me it is not a friendly game, it is really important.
"The way that we had been preparing for the game, the team wanted to play. Sometimes it is difficult to go and play against a team from a lower division with the right mentality, but in the second half we played with the right attitude.
"Even when we lost Dani [Ayala], we were really consistent, everyone was committed, everybody was fighting for every ball. That's the main thing for me.
"The image we were showing [in the first half] wasn't right, it wasn't the same way that we had been preparing for the game in the week, the way I wanted to play."
Liverpool example spurs on Boro
Before the game, a tie with one of the Championship's play-off favourites looked like a tricky one and, in the aftermath of Liverpool's draw with Plymouth Argyle, Karanka said his side were well aware what was expected of them.
"It was the type of game where [Wednesday] know that they are in the Championship but they want to come here and show that they can compete against us," he said.
"In every single game, you have to give 100%. Today, Liverpool drew [against Plymouth Argyle], which is a good example that you have to go to the pitch in every game and give 150%."
Captain Leadbitter continues to please Karanka
Karanka claimed that Ayala's red card, awarded for a heavy foul on Wednesday's Fernando Forestieri, had come as a shock at the time but that he was yet to watch a replay of the incident in the aftermath of the game.
The 43-year-old revealed that he had identified Wednesday reserve keeper Joe Wildsmith as a potential weakness in the Owls' line-up ahead of the game, and he was pleased with Leadbitter's decision to test him with an unexpected free kick towards the keeper's side of the net.
"I think it was a surprise to everybody, but I was told that it was a red card. I haven't watched [a replay], so I don't know if we will appeal yet.
"It is another thing I told them at half time, that they were playing with their third keeper and we didn't know if he was good or bad. Grant's free-kick was intelligent, and after that goal we were much better.
"One or two minutes later Dani was sent off and I needed to change a few things, but with ten men our commitment was very good."