Two cracking strikes from range by midfield duo Ben Pringle and Lee Frecklington sealed the win before half time, either side of Rhys McCabe's 30 yard free-kick for the visitors.
Wednesday later had sub Jermaine Johnson sent off for aiming a headbutt at livewire Pringle, as the Millers booked their place in Thursdays second round draw.
Gaffer Evans told themillers.co.uk after the game: “It was a fantastic performance from start to finish. Some won’t like me for saying it but we’ve been a really good team with an excellent manager in Dave Jones. They’re a talented side but we were on top of our game on the pitch and in the stand. They (the fans) have been absolutely fantastic and it was a great advert for cup football.
"These are the type of cup games that the supporters need and desire like me and what about the atmosphere that they created tonight. For 15 months the backing has been incredible.
“The staff and players are inside debating who scored the best goal, because they were both very special and you’ll struggle to see better. It doesn’t matter to me whether it’s a tap in, and a sharp Tubbs would have got two goals, Revs went close as well and maybe if anything, Wednesday are fortunate to only lose by the odd goal.
"They lacked a bit of discipline and when they went down to 10 men we saw the game out well.”
The second round draw is live on Sky Sports News at 12.30pm tomorrow.