The 35-year-old, who had been in interim charge since Phillip Cocu was sacked on 14 November, has now also officially retired as a player.
Rooney has overseen nine games so far, winning three and drawing four.
"The opportunity to follow Brian Clough, Jim Smith, Frank Lampard and Phillip Cocu is an honour," he said.
"I knew instinctively Derby County was the place for me."
Liam Rosenior takes up the role of assistant manager, with former England boss Steve McClaren continuing as technical director and advisor to the board of directors.
Shay Given will become first-team coach and Justin Walker will remain as first-team development coach.
The Rams are third from bottom in the Championship, level on points with fourth-from-bottom Sheffield Wednesday.
A takeover for the club is expected to go through this week, with a deal between current owner Mel Morris and the Deventio Holdings Group having been agreed in November.
Rams chief executive Stephen Pearce said in an interview with BBC Radio Derby on Thursday that their were no problems with the takeover, despite the delays meaning players have not been paid their December wages.
"Our recent upturn in results under Wayne was married together with some positive performances, notably the 2-0 home win over Swansea City and the 4-0 victory at Birmingham City," said Pearce.
"During that nine-game run we also dramatically improved their defensive record and registered five clean sheets in the process, while in the attacking third we became more effective and ruthless too.
"Those foundations have provided a platform for the club to build on in the second half of the season."
Rooney made his professional debut for boyhood club Everton in August 2002 aged just 16 and became the Premier League's youngest scorer with a superb long-range goal against Arsenal before his 17th birthday.
After a strong Euro 2004 he moved to Manchester United for £27m, then a world record fee for a teenager.
During 13 years with United he won the Premier League five times, the Champions League, the FA Cup and three League Cups.
His time with England was less successful in terms of team honours, although he did break Sir Bobby Charlton's long-standing record of 49 goals before retiring from international football in August 2017.
He made a farewell appearance for the Three Lions against the United States in a friendly in November 2018 to finish with 53 goals in 120 appearances.
After a second stint at Everton and a spell with American side DC United, Rooney joined Derby in January 2020 as a player-coach on an initial 18-month contract.
He retires as the second-highest goalscorer in Premier League history, with 208 goals.
Ed Dawes, BBC Radio Derby's Rams commentator
Wayne Rooney's presence at Derby County was felt on that hot August evening in 2019 when Phillip Cocu won his first match as manager at Huddersfield, a result overshadowed by the announcement of his signing.
Rooney's ambition to become a manager was there for all to see when chairman Mel Morris afforded him the opportunity to be a player-coach on arrival in January. He in fact arrived a few months before that but was unable to play, and stayed low key, observing from the sidelines.
A year ago this month he made an instant impact to Derby's fortunes on the field. Players who were underachieving and perhaps found the grind of the Championship a little hard to handle, were taken up a notch by his presence.
Some would say Rooney saved the Rams' season, but this term he struggled on the field and so did Derby.
I am told it was written into his contract that he would have a chance to take control one day and he has already shown in his nine games in interim charge that he can get the squad playing in his image. Gone is the side-to-side, slow build-up possession game, it is a better product to watch.
The people around him have good pedigree in the game. Shay Given, Liam Rosenior, Justin Walker and Jason Pearcey have experience at all levels - but his relationship with Steve McClaren will be the most important of all.
Derby fans have been calling out for a positive piece of news. Rooney's appointment is the first duck in a row with the takeover expected to be completed any time now and then Championship survival is the hope.