Rooney, the former Manchester United and England striker, was appointed Birmingham manager on Wednesday.
“Two wins on the bounce is good for us, it’s all about results and momentum and we can use this as a real springboard.”
Liverpool needed penalty kicks to overcome League One Derby County after a goalless 90 minutes ended with the titleholders securing their place in the fourth round.
Rooney is leaving troubled Derby County, who relegated to League One last season, one year to the end of his contract.
Derby County chief executive officer Stephen Pearce said Rooney was the perfect man for the job.
