Heaven has penned a deal to 2029 with United as he leaves Arsenal for an undisclosed compensation fee.
The England U19 international made his senior debut for the Gunners in October.
"I am incredibly proud to join Manchester United. I'm grateful to everyone who helped make this dream a reality," he said.
"There is so much that I want to achieve in the game, I'll be giving everything to continue my development and become the best player that I can be."
Manchester United technical director Jason Wilcox said: "We are delighted that Ayden has joined Manchester United. He is already an extremely talented defender who is ready to join our first-team squad in order to maximise his development.
"This club has a phenomenal track record of supporting young players and we believe that Ayden is in the perfect place to reach his excellent potential."