The club, which is renowned as the home of golf, voted last September to end its male-only membership rule, breaking with a 260-year tradition.
Princess Anne, who is also a member of the International OIympic Committee, and Sweden's Sorenstam, were granted honorary membership status.
The other five to be honored, Britons Laura Davies and Belle Robertson, Americans Renee Powell and Louise Suggs and Lally Segard of France, have played and enjoyed success at the highest levels of competitive golf.
"This is a historic day for golf," George Macgregor, the captain of The Royal and Ancient Golf Club (R&A), told its official website.
"They are extremely worthy additions to our roll of honorary members and will become ambassadors for the club as they have been for the sport of golf throughout their careers."
Princess Anne, the only daughter of reigning monarch Queen Elizabeth II, joins three other members of the British royal family in becoming an honorary member of the R&A.
The club, founded in 1754, organizes golf's oldest major, the British Open, and also administers the rules of the sport from its famous headquarters in St Andrews in Scotland.
It was also announced Tuesday that an unspecified number of lay women members had also been granted membership with more set to follow in the coming months.
The R&A had come under increasing pressure to admit women members, with former British prime minister Gordon Brown among leading public figures to voice their concerns on discriminatory grounds.
It eventually followed the lead set by the exclusive Augusta National Golf Club in the United States, which admitted its first two female members, former Secretary of State Condoleezza Rice
and business leader Darla Moore in August 2012.
In the R&A vote last year, 85% of its members backed the move to admit women members in an historic ballot.