First Minister Nicola Sturgeon hailed the announcement as "fantastic news" for the capital's economy and shows that Scotland is open for business.
It came as new figures showed that inward investment in Scotland secured almost 8,000 jobs in Scotland last year, a 10% hike on the previous period.
Computershare pocketed a £2 million grant from national economic development agency Scottish Enterprise to fund the expansion.
Ms Sturgeon said: “This announcement, with the creation of 300 highly skilled technology jobs and investment in the city centre, is fantastic news for Edinburgh’s economy.
“Scotland is open for business and continues to be a very attractive location for investment, as evidenced by the recent EY Attractiveness survey, which noted that Scotland was the top UK location for foreign direct investment outside London for the fifth consecutive year.
“Together with the inward investment figures published by SDI, this offers further evidence that we have the skills and expertise to attract and retain global companies like Computershare.”
The expansion plans were announced on the same day Scottish Development International annual results were published, which showed 7,839 jobs were secured in Scotland through new and existing investors.
Stuart Irving, global President and CEO of Computershare said: “As a truly international capital city, Edinburgh has a bright future and is a natural home for a global company. As a growing business we need the skills and hard work we see on offer in this city.
“We are grateful to the Scottish Government, Scottish Enterprise and Scottish Development International for helping us with our plans and are looking forward to our continued partnership.”