Parts of the book were widely leaked but its contents have continued to fill airtime and newspaper and online pages.
Speaking about the sales, president and publisher of the Random House Group Gina Centrello said in a statement that "Spare" was far more than a celebrity memoir.
"Vulnerable and heartfelt, brave and intimate, 'Spare' is the story of someone we may have thought we already knew, but now we can truly come to understand Prince Harry through his own words," she said.
"Looking at these extraordinary first day sales, readers clearly agree, Spare is a book that demands to be read, and it is a book we are proud to publish," she added.
The book is one of a number of lucrative contracts struck by Harry and his American wife Meghan cashing in on their royal connections.
The couple dramatically quit royal life and moved to California in 2020 and have complained bitterly about their treatment in the UK.
A YouGov survey published in The Times found that only 24 per cent of people now have a positive view of the prince -- down from 80 per cent a decade ago -- with 68 per cent critical.
Among over-65s, Harry and Meghan's popularity ratings were even worse than those of the disgraced Prince Andrew, the survey found.
"Spare" was published in a total of 16 languages worldwide and released simultaneously in print and digital formats in North America by Random House US and Random House Canada, and in the United Kingdom by Transworld.
An unabridged audio edition of the book, read by Harry, was also published by Penguin Random House Audio.
The first US printing of Spare was two million copies, however the book has now gone back to press for additional copies to meet demand, according to Penguin Random House.
Bookshops in the UK opened early on Tuesday to offer Harry's book although buyers were sparse after the extensive leaks ahead of publication.
The blockbuster publication comes as the royal family prepares for the coronation of Harry's father King Charles III on 6 May.
The royal family has not commented on the book.