The former Manchester United star agreed a three-and-a-half year contract with the Championship club on Wednesday after leaving Major League Soccer side DC United at the weekend.
He replaces John Eustace, who was sacked despite guiding Birmingham to sixth place in the early weeks of the season.
Rooney, who was previously in charge at Derby from 2020 to 2022, told a press conference at St Andrew's: "The goal, for the club, for myself, is to get this club back to the Premier League, of course.
"I think there's a lot of work to be done throughout the football club. The Premier League is where we want to get to.
"It's an ambition of mine, one of the club's, and we're putting everything in place to make sure we do that in the near future."
He said he had wanted to return to English football after his stint in the United States.
"I've had opportunities over the last four, six weeks at other clubs as well, to do that," he said.
"But I think since speaking to Birmingham and seeing the ambitions of the club, where they want to go, where they want to get to, it excited me.
"I want to be successful. It's clear this club wants to be successful, and everything we spoke about really was very similar. It was a really easy decision once I'd spoken to them."
The former England captain was unable to stop Derby from being relegated to the third tier in 2022 after the club were deducted 21 points for entering administration and financial breaches.
He also failed to take DC United into the MLS play-offs this season.
Rooney was a teenage sensation as a player with boyhood club Everton before he joined Alex Ferguson's Manchester United in 2004.
He became United's record goalscorer and won five Premier League titles and the Champions League along with other silverware at the club.
Birmingham currently occupy the final play-off place in the Championship, 10 points behind Ipswich, who are second in the table, in the second of the automatic promotion spots.
US-based Shelby Companies Limited completed a takeover of Birmingham in July, with seven-time Super Bowl champion Tom Brady becoming a minority owner in the club.
