The Millers continued their fine festive form with goals by David Ball and captain Lee Frecklington inside the opening 10 minutes.
Will Vaulks then fired in a superb 83rd-minute strike to make it 10 points from 12 for Rotherham, sending them up to eighth in League One.
Warne said: “We pressed high up the pitch, tried to play with high energy and give ourselves chances to score, and for the first 15 minutes we were virtually unplayable.
“Most of Bury’s points have been at home and the fans are normally loud on Boxing Day so if they got an early goal they’d have got right behind their team.
“We wanted to start well and we managed to quieten the crowd down. It was good to get two goals when we were on top and the third late on was the cherry on the Christmas cake.
“There were some really good performances so I get on the bus back to Yorkshire very happy.
“The lads had a bit of a wobble in terms of points collection but the lads are now in a good place and are buying into what we’re trying to do.”
The Millers went in front after five minutes as Tom Aldred cleared Ryan Williams’ cut-back as far as Ball, who curled inside the far post from 12 yards.
Anthony Forde’s corner then found Frecklington and the Rotherham midfielder powered in a free header.
Williams’ low strike was turned wide while Jay O’Shea and Danny Mayor fired wide as Bury improved slightly after the break.
But Vaulks fired in from the corner of the box to leave Rotherham two points off the play-off places while Bury are now nine points from safety.
Their manager Chris Lucketti said: “I’m really disappointed. The first half was totally unacceptable.
“The fans were having a go at the players and rightly so. I’m fully supportive of that and I told the players that in no uncertain terms at half-time.
“A lack of effort, lack of commitment and lack of desire was apparent for all to see, and you can see why the club is where it is so far this season.
“I laid into them at half-time and we saw a reaction second half. It was a totally different performance, but it was too late.
“The first two goals were avoidable but we’re saying that every week – and I’m sure it was said before I came in.
“You’ve got to do the basics right and we’re not doing that in terms of defending our box. The next game can’t come quick enough. I want to see a reaction from the players on Saturday.”
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