The Reds had already conceded first to Everton, Fulham, Southampton, Liverpool and Tottenham this term when Pantelis Kapetanos shot Cluj ahead from the Romanians' first truly dangerous attack.
Rooney, who was excellent again in a deeper role, acknowledges that it is a dangerous habit that needs fixing in order to compete for silverware.
"Obviously it's not nice," he told MUTV. "It's something we need to make sure we stop doing. We can't keep affording to let teams get a headstart on us, especially in the Champions League when it's very difficult to win the game anyway.
"We have to make sure we're more solid than we have been and hopefully we can be in future games."
Cluj were difficult opponents and the surface did not aid the Reds' cause but Rooney was pleased to make it six points out of six in the competition.
"It was quite tough at times but, thankfully, we've got the three points and that's the main thing," he stated.
"I played a bit deeper tonight and had some good moments [with van Persie]. Robin has got two goals tonight and we're delighted. I didn't know the first one was off his shoulder until he said it at half time but as long as it goes in, it doesn't matter.
"It was a great finish for the second goal. I was right behind it. I could see it was a fantastic finish.