North End slipped a goal behind after just a couple of minutes, as Marco Stiepermann played Adam Idah and to get his first of three, all assisted by Stiepermann.
Another goal from Onel Hernandez made it four for the Canaries, with Josh Harrop and Billy Bodin scoring for PNE in a resurgent second half, but it was too little too late in the end, scoring when 3-0 and 4-1 down.
But for Norwich boss Farke, the spirit of North End caught his eye and he thought they also lacked a bit of luck as they fell three goals down, with two errors from Connor Ripley in goal.
He said: "It’s a bit unlucky when it’s early in the game and you’re 2-0 or 3-0 down in the game. It’s difficult, especially because they also had many changes in the team like we had.
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"To keep the belief is difficult but I never had the feeling that they would give up. I like the mentality and the attitude of this team.
"Even when they got the first goal you get the feeling that okay, they want the second goal and to try and push us.
"We had several good spells where we played some fantastic football but there was also a few spells when we had to show some steel because Preston tried to push and push.
"I like that mentality a lot, of the team, with this mentality and fighting spirit I’m pretty sure they will win many, many points at Championship.
"I liked their spirit, especially in the second half when they came back."
The German was aware of PNE's strength at home this season, and was happy to steer his side into the fourth round for the first time in seven years.
"We are pleased that we are through to the next round. Someone told me it’s the first time in seven years that we have been able to manage it, so it’s quite good.
"I’m really pleased today because it is a tough draw for us. It’s our fifth game in 14 days and then an away game at a side who are fighting for promotion, with the best home record in the Championship.
"Then to play here was difficult but if I’m being honest, I never had a feeling like we would lose this game.
"From the first to the last second I thought it was a really professional and concentrated and well-rounded performance. I just have compliments today for the lads, I think they were brilliant and we deserve to be through to the next round."