Blackburn had to withstand long spells of pressure in Saturday's match as Colin Kazim-Richards scored the crucial goal to fire the Championship side into the quarter-finals.
Appleton was delighted with the effort shown by his players but says they must keep their feet on the ground and produce a similar performance in the league next time out.
He enthused: "I have said it for a while but it is about hard work, organisation and team effort. I thought the lads were fantastic.
"We will enjoy that and then focus on Hull City in the league on Tuesday night. Matches like that have to be great for the confidence.
"The fans were great and I'm sure they will have a good night."
Appleton was particularly pleased with the efforts of Scott Dann and Grant Hanley in defence, as Blackburn were able to take advantage of the fact that Arsenal left out some of their senior players ahead of their UEFA Champions League clash with Bayern Munich on Tuesday.
"To come to Arsenal and get through a cup tie is one thing, but to keep a clean sheet and defend as well as we did is another, and that is testament to our players. We have two man-mountains in the centre of defence," he said.
"We had a feeling they might rest a few of their players because they have a big game on Tuesday in the Champions League.
"However, regardless of what side they were going to put out, we knew we would concede a lot of possession at times, but we are happy with that.
"We have a mentality now where we are hard to beat, and are resilient.
"It has been almost getting into the heads of the players to take pride in keeping clean sheets and enjoy defending, which people do not like doing at times, we also have a bit of quality as well."