Skipper Ryan Wintle was dismissed after just 21 minutes for felling the goal-bound Harry Gray.

The early blow failed to derail the rampant visitors.

They swiftly took control of the game, despite the man disadvantage, scored three emphatic goals and kept hold of top spot.

Barry-Murphy said: “I was very pleased with the players’ mentality and how they played the game.

“When Ryan was sent off we had to show clear signs of the team we want to be and I am really grateful that the players did that for us.

“We focused on ourselves but today, from the first minute, I had a good feeling with the team.

“We looked really like ourselves and I was very confident at all stages.

“I was really pleased with the character and mentality of the team.

“To score the goals and create the chances that we did is testament to the players and the work they are doing.”

Barry-Murphy shrugged off the suggestion that this three points, under the difficult circumstances, was a strong message sent out to their title rivals.

He replied: “I think it’s a very important message to ourselves.

“We weren’t pleased with how we played at Burton last week and we wanted to better that.”

Ollie Tanner turned provider on two occasions, either side of the break, to put Cardiff in the ascendancy.

The first goal came from Tanner’s cut-back to Omari Kellyman who guided the ball past Cameron Dawson clinically.

The advantage was doubled when Tanner picked out Chris Willock in acres of space and he was able to take his time and slot home.

Substitute Isaak Davies rounded off the scoring in added on time as he burst forward on the counter-attack and lashed beyond Dawson.

Rotherham remain deep in relegation trouble and are two points from safety.

Manager Matt Hamshaw said: “Cardiff are a good team and play some good football.

“I thought when they went to 10 men it was an opportunity for us but we played into their hands.

“I just felt we got too complacent as a group and I reiterated that at half time and we didn’t get much better.

“We can’t have Jekyll and Hyde performances.

“The battling qualities we had last week weren’t there this week.

“All in all, everybody performed miles below the standard we have had and that is a frustration.

“I think Cardiff are the best team in the league and will get promoted.

“They stopped us doing anything but we underperformed. I didn’t see that coming.”