The promotion-chasing Blades led through Gary Madine’s 51st-minute strike but ended up dropping points for the second successive Sky Bet Championship game.
John Egan was sent off for a handball on the line and while Ben Marshall missed the resulting penalty, Millwall found an equaliser through Jake Cooper’s diving header deep in stoppage time.
Wilder said: “They’re tough opposition to play against and they obviously set themselves up at the start to play on the counter-attack. It was difficult for us to get the initiative with everyone behind the ball.
“When they go a goal down they have to show more ambition, but we just went backwards instead of trusting the shape. We went backwards in thought and in play and paid the price.
“After the goal they deserved a point, but we never showed enough ambition to go on and score the second, which is uncharacteristic from our team.
“They missed a penalty, but we didn’t just deserve a point. We have deserved more in other games like today, but that is how football is.
“We played safe and didn’t go for the jugular. We played backwards and square and sat in, so I’m not surprised they got back into the game and deservedly so.
“They had a drive to get back into the game and that’s one of their characteristics. It’s stinging that we didn’t get those extra two points.”