A half-time dressing down from Sean Dyche inspired a second half revival at Middlesbrough
Burnley were not at the races in the first half at the Riverside Stadium but went in level as Rouwen Hennings cancelled out Diego Fabbrini's opener.
Stephen Ward went on to score the winner as Burnley progressed to the fourth round of the FA Cup for the first time since 2011.
Dyche said: "Second half pleased me, first half I had a quiet word at half-time, it's fair to say.
"They are a good side, it's a tough place to come, but we didn't play our game, and we agree we would.
"Apart from a flash of superb play for our goal...they want you to play your way and let you play and wait for the chance to take it from you and then play their game.
"But second half we made it way more uncomfortable for them and that was on show.
"Overall, after Tom Heaton playing very well in the first half - they had two or three really good chances - second half they had nowhere near as many, and I was really pleased with that tactically from the team."
Dyche felt there was too much respect for Championship leaders Boro in the first period: "I think sometimes there's a bit of respect and it goes onto the pitch. You don't want it to, respect is before you play, and when the whistle blows, it goes out of the window.
"There was a bit too much respect, and you can imagine that, they're going great, conceded two here before today.
"Footballers aren't daft, they think it's a tough challenge here, and there was that feeling.
"Second half, after a short reminder, we came out being us, and the mentality was excellent, playing our game, to bring them out of their game."
Hennings levelled right on half-time, the turning point, and he added: "The goal gives you something when you've not played well. I don't think they were brilliant, it was more a case of us not playing well.
"We just weren't on top of our game, but second half it was obvious to everyone in the stadium that we were, the first 15 minutes in particular.
"It's unusual teams come here and dominate like we did the first 15 minutes of the second half.
"They had a little spell, and then we finished organised and strong."