The news was announced on Thursday after three-quarters of the club's 2,400 members took part in the ballot.
The final result showed 85 per cent in favour of allowing women members, breaking a 260-year history of male-only membership.
"This is a very important and positive day in the history of the Royal and Ancient Golf Club," said chief executive Peter Dawson.
"The R&A has served the sport of golf well for 260 years and I am confident that the club will continue to do so in future with the support of all its members, both women and men."
Sports minister Helen Grant added: "I am pleased that the members of the Royal and Ancient Golf Club of St Andrews have voted in favour of admitting women members.
"This is positive news for the sport and I hope we will now see other golf clubs that still have outdated same sex policies follow suit.
"With golf in the next Olympics there is a huge opportunity for the sport to grow and this sends out the right inclusive message that golf is for everyone."
The Royal and Ancient Golf Club is based at the historic course of St Andrews in Scotland, where The Open Championship will be held next year.