The victory at the Reebok Stadium was achieved only three days after Gary Megson's sacking.
An own goal from Moses Swaibu set the hosts on their way, with Chung-Yong Lee, Gary Cahill and substitute Mark Davies also on target.
Evans, working alongside coaches Steve Wigley and Alan Cork, said: "Given the circumstances of the last few days, it was very important the players were focused.
"It is very difficult when the attention of the football world and the media are on you.
"Steve Wigley, Alan Cork and the staff worked very hard, especially in regard to the mindset of the players. We did a professional job and are pleased to be in the hat."
Evans picked a strong side for this third-round tie, and added: "We had to pick a side that was going to get a result.
"People say the FA Cup has lost its magic but, as a traditional football man, I love it. I feel we have a duty of care to the competition."