The winger was dismissed for a headbutt after being on the pitch for just 18 minutes as the Owls were knocked out of the Capital One Cup by their League One South Yorkshire rivals.
Jones said: "It was just stupidity , you can't headbutt somebody, he's done it before and it's not acceptable for this football club . He's been told I can't defend it and neither can JJ.
"We're trying to get back into the game, it's hard enough with 11 men and going down to 10 makes it more difficult. He'll be punished and rightly so because I lose him for three games.
"What cost me the game is we didn't defend two long throws into the box. We didn't come here and compete hard enough and do enough around their goal area full stop.
"It's disappointing, we knew they were going to put balls into the box, kick them in or throw them in. I was disappointed we didn't get a grip of the game from the word go but we knew it was going to be difficult. We know the way they play, playing for free-kicks and throw-ins, and they are going to put it into the box.
"I thought there was a bit of a panic with us for clearances . It's disappointing because we know we are better than that."
The home side went in front when Kari Arnason's long throw was cleared to the unmarked Ben Pringle and the midfielder lashed a rasping 20-yard effort past Chris Kirkland with just 10 minutes gone.
With a rare break forward Wednesday won a free-kick 30 yards out which was struck low with precision by Rhys McCabe to level after 23 minutes.
But another Arnason throw-in fell to Lee Frecklington, who restored Rotherham's lead with a sweet strike after 38 minutes - his third goal in the opening two games of the season.
Rotherham hung on to their lead in the second half as Johnson was duly sent off after 76 minutes.
Delighted Rotherham boss Steve Evans said: "It was a great win for us against a great football club. We always believed we could win but we knew we would have to play well.
"We did our homework and we had a game plan. We have talented players and we scored with two great strikes . It was a magnificent cup tie and it was a great achievement for this football club to win at home for the first time against Sheffield Wednesday for 37 years.
"We knew how tough it would be. They are a great club with a great manager and some talented players and it shows you how well we have had to play to get a result "
He agreed with Jones over Johnson's red card, adding: "He is such a special, talented player but talented players live on the edge and it's a shame."