Plans include renovating the hidden theatre, dating back to 1875, to use for plays, films, comedy and music.
Material from the BBC archives will be shown in revamped TV studios, now derelict, where the broadcaster first launched a regular television service almost 80 years ago.
The plans will be finished by 2018.
Both the theatre and the studios, which have been derelict for 35 years, are in the eastern end of the Victorian palace.
The section will be refurbished and repaired, Alexandra Palace said, along with a new entrance hall.
The theatre will be able to accommodate 1,300 people.
More than 2,000 local residents were involved in a six-week consultation leading up to the decision, Alexandra Palace said.
Chief executive of the Alexandra Palace and Park Charitable Palace Trust Duncan Wilson said he was "thrilled" with the announcement.
Mr Wilson said there would be further challenges ahead, such as securing more funding and preparing the palace.
Chief Executive of Heritage Lottery Fund Carole Souter said: "Alexandra Palace has played a pivotal role in the development of popular entertainment - culminating in the birth of British television."
The total cost of the project is £26.7m.
The trust will raise an extra £1m before the project starts in 2016, it said.
Alexandra Palace has been given £18.8m from the Heritage Lottery Fund to restore a forgotten part of the building.