He will replace Tony Hall, who was last year named as the BBC’s new director general.
A statement from the ROH said Beard would take up his post in time for the start of the 2013/14 season, which begins in September. Sally O’Neill, finance director, will act as interim chief executive until he arrives.
Beard, who is 49 and was awarded a CBE in this year’s New Year Honours, has been at the Tate since 1994 and has been deputy director since 2002.
His salary will be £250,000, which is significantly less than the £392,361 Hall was paid for the year to the end of August 2011, according to its annual accounts.
Simon Robey, chair of the ROH, said Beard had a passion for the work it did. "I am confident he will forge excellent partnerships with our artistic leadership and our executive team, and that they will, together, lead the Opera House to still greater heights," he said.
Beard said: "To have the opportunity to build on Tony Hall’s achievements and strengthen the Royal Opera House’s reputation as one of the world’s leading artistic and creative forces is hugely exciting."
In the financial year 2010/11, the Royal Opera House had a turnover of £109.5m.
Nicholas Serota, director of the Tate, said: "Alex has made a massive contribution to the development and success of the Tate since 1994. His quick intelligence and sense of enterprise has allowed us to grasp opportunities as they have appeared, and I know that he will be a real asset to the opera house."