Goals from Jack O'Connell and Mark Duffy sent United back into the automatic promotion places with a 2-0 win over the Millers in the day's early kick-off, and the turning point of the game was arguably the early decision to send off Vaulks for a reckless lunge on George Baldock.
Chris Wilder, Warne's opposite number, had no complaints about the decision but Warne said: "I was a long way from the incident and I haven't seen it back, but because Will gave the ball away just before I think the ref's thinking 'oh God, he's lost his head and there's going to be a tackle here'.
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"We've not had many reds and we're not a dirty team, Will's a really competitive player which is why he's so good for us.
"I don't think there was any intent to hurt the lad and I thought at the time it was a strong yellow, but I knew we were in trouble when the ref reached for his back pocket."
Marek Rodak, the Millers goalkeeper, pulled off a string of fine saves to keep the score at 2-0, with a stand-out stop to deny Enda Stevens' towering header.
Warne added: "If I'm wrong I apologise, but from where I was stood I thought it was on the edge of harsh.
"We tried to stay in the game for 30 minutes, everyone knows how United play - they try and outnumber you out wide.
"It asked a lot of our lads physically and my 'keeper pulled off some great saves but to be fair, that's why I pay him.
"On another day a set-piece might have got us back into it, but United were more than deserving of the win."