She said she was "so proud and excited" about her new role, describing it as "a dream come true".
Terry, who has performed at the Globe in As You Like It, A Midsummer Night's Dream and Love's Labour's Lost, will take over from Emma Rice in April 2018.
Terry won an Olivier award for her performance in Tribes at the Royal Court in 2010.
She also wrote and starred in the Sky One series The Cafe, with Ralf Little.
Neil Constable, chief executive of Shakespeare's Globe, said: "Audiences have loved her sparkling and intelligent performances on our stage over the years - now they will have the opportunity to see her fresh artistic vision come to life as she moves into a new phase of her impressive career."
Terry most recently starred as the king in Henry V at Regent's Park Open Air Theatre, and as Grace in Katie Mitchell's production of Cleansed for the National Theatre.
The actor-manager is a role as old as the theatre itself, and is a tried and tested practice at the Globe - with Mark Rylance combining the roles with considerable success during his stint as its artistic director.
There is no obvious reason why an intelligent and versatile actress such as Michelle Terry, who is immersed in Shakespeare and the Globe, shouldn't make a success of it too.
This is as long as she is free to pursue her creative vision without unwelcome intrusion from a board who made a complete hash of the Rice era.
The news comes following the controversial announcement of Rice's departure. She is leaving after the Globe's board decided her methods were not authentic enough.
Rice took charge of the London theatre in January 2016 but has come in for fierce criticism, including for her use of sound and lighting technology.
Earlier this year Rice wrote an open letter to her successor, in which she claimed the board had sought to impose "a new set of rules that I did not sign up to and could not stand by".
Rice is now launching a new theatre company at London's Old Vic.
The Globe, which opened in 1997, is a reconstruction of a Shakespearean theatre on London's Southbank.
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Actress and writer Michelle Terry has been named as the incoming director of Shakespeare's Globe theatre.