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 has already ruled himself out of the running to replace Megson, with Burnley's Owen Coyle emerging as a strong favourite.
And former Wolves academy chief Evans insists the club can progress under new leadership.
He said: "I believe there is a real chance of ongoing success. The manager who comes in will have reason to feel positive about the future as we have solid foundations.
"We will come in on Monday, ready to work hard. It is about hard work, it is about focus. It is incumbent on us to do the best we can."
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."
Lincoln manager Chris Sutton saw his team fold after a goalless first half, but insisted: "The effort and commitment of the players was fantastic.
"We played some decent stuff and did not feel they were much of threat early on. But then we conceded an own goal and were all at sea for the second.
"We knew it was going to be very difficult after that but we did not disgrace ourselves in any way.
"This is mainly a young side and, despite the scoreline, the players can be proud of their efforts."