The Millers claimed their second Sheffield scalp of the season and moved into fourth place in the League One table with their first home league win; Evans admitting he "got it wrong" in the first half.
The boss said he apologised to his team at the break, changed things around and reaped the rewards.
Jose Baxter's first goal for the Blades gave the visitors a half-time lead, yet after the interval there was no stopping the home side.
"The difference between the first half and the second half was that I got it wrong in the first half," said Evans afterwards.
"In the first half we just weren't in the game and as a manager I got it wrong. I apologised and in the second half I got it right, but I should be getting it right from the start.
"We tried to suppress some of their players in the first half, but that went out of the window after 10 or 15 minutes when our skipper at the back was injured.
"At half time we made a change because we had to get more support to the front boy and the change allowed us to do that."
Evans added: "We have made a steady start to life in League One; a steady start rather that a great start. There are some very, very good teams in this league and we have just beaten one of them."
Blades boss Davie Weir was very disappointed after the game, saying: "It was the manner of the collapse in the second half that hurts the most.
"After being comfortable in the first half the second half was very, very disappointing. The manner of the goals makes it doubly disappointing.
"I think we contributed in a big way to our downfall, there's no doubt about that. The level of the goals we conceded and the level of the mistakes. It is going to be difficult to win a game of football when you commit that level of mistake.
"We were on top in the first half, but a game of football doesn't last 45 minutes, it lasts 90 minutes. It was the reaction to their first goal that was the worst thing about the game. We have got to be stronger than that and obviously that is another disappointment.
"We did well in the first half and got ourselves in a good position, but you get nothing for that."
- Sports & Recreation