The Reds opened the scoring through Jacob Brown then, after the League Two hosts levelled, Conor Chaplin and Luke Thomas secured a place in the fourth round or the first time since 2013.
Rose said: “For us it’s important to keep our run of good performances and results going and our confidence up. It was important not to follow up the Derby defeat with another defeat. That would have really dented our confidence so this was very important for us.
“We started the game very well and had a gameplan which worked very well for 15 minutes but we dropped after that and invited them back into the game with too many long balls. There was a spell for five minutes in the second half when they equalised but we rode that well and, in the end, it was quite professional and a job well done. It’s good for confidence and team spirit.
“Crewe are a good team. If you let them play, they will cause you problems. What we managed to do was stifle that. I don’t think anyone wanted a replay, with the amount of games you get in the Championship. We had very good support in the away end and the players welcome that. It’s important the players thank the fans after every game because they are the heartbeat of the club.”
Barnsley now want a Premier League giant in the next round. Rose said: “Everyone wants a top team. The chances of winning the FA Cup outside the top clubs are very slim so the players will want a big team which is their chance to shine.
“Gerhard spoke to the players about their motivation for today. The common fact from the players was that they want a good cup run with a game against a big team so they can test themselves. I am sure tomorrow night the fans will be watching the draw on TV and hoping to draw a big team.”
Thomas scored the final goal in injury-time, into an empty net after the home goalkeeper went up for a free-kick. Rose said: “Luke came alive all of a sudden and seem to go from nought to 100 in five seconds. That’s cup football. It could have gone the other way and they could have scored.”
Thomas said: “I saw it fell to Kenny (Dougall) on the edge of the box and I knew he was going to clear it so I just got my head down and ran. There was a discussion in the dressing room about whether it was going in but it definitely wasn’t. I needed it because I haven’t scored for a while. It was brilliant to do it in front of a packed away end as well.
“It’s great to get through to the next round. These cup ties are never easy and we had a meeting yesterday to make sure every player was up for it. In the end, it was a professional performance. We had to be wary of what Crewe can offer because they are a good footballing side. We have a lot of character in the squad now, especially after the new manager has come in.
“One of the motivations and why the FA Cup is so good is to get a big club in the next round. That’s why all the lads wanted to get through so bad. I would like Arsenal because that’s the team I supported when I grew up. Them at Oakwell or the Emirates would be lovely.”
Barnsley will now have some time off after five games in 15 days before welcoming local rivals Huddersfield Town in the Championship as they attempt to cut the three-point gap to safety.
Rose said: “We have had a good run of six or seven games but, in the last two, the schedule and the effort we put in to get back from an eight point deficit showed a bit in our legs. Now we have a week to regather and go again. We have a few days off for the boys to get their legs back then we go again for Huddersfield which is a very big game.”
Rose said Clarke Oduor went off with a tight hamstring but it was only a precaution.