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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He was charged after catching West Ham's Winston Reid with his left arm as they waited for a free-kick in Saturday's 0-0 draw at Old Trafford.
The incident was not seen by the match officials but was caught on video.
Schweinsteiger, 31, will miss games against Bournemouth, Norwich and Stoke, and return in time to play Chelsea.
Referee Mark Clattenburg spoke to both players after the incident, but did not mention it in his post-match report.
A panel of three former elite referees each reviewed the video footage independently of one another and unanimously decided it was a sending-off offence.
"He knows what he's done - he's elbowed me in the face," Reid said following the match.
West Ham manager Slaven Bilic said it was "a clear red card - a straight red without any doubts".
He added: "It was a moment of madness. He deserved to be sent off, but the referee told me he didn't see it. Fair enough, I believe it. It was an elbow."
Germany captain Schweinsteiger, who was part of their 2014 World Cup-winning side, joined United from Bayern Munich during the summer and has made 24 appearances for the club, scoring one goal.

The Dow Jones industrial average ended 11.23 points ahead at 19,945.
Investment bank Goldman Sachs led the risers, up 0.24%. The sportswear giant Nike led the fallers, down 1.2%.
The technology-heavy Nasdaq jumped 24.75 points to 5,487 while the S&P 500 added 5.09 points to 2,268.
Oil prices increased ahead of a deal to cut production.
Brent crude gained 1.7% to $56.09 per barrel while US crude also rose 1.7% to $53.90 a barrel.
The Opec cartel of oil producing nations have agreed to reduce output from 1 January 2017 by 1.2m barrels a day to a daily 32.5m barrels for at least six months.
Actress and writer Michelle Terry has been named as the incoming director of Shakespeare's Globe theatre.
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Manchester United midfielder Bastian Schweinsteiger will be banned for three matches after accepting a Football Association charge of violent conduct.
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Wall Street closed higher on Tuesday but surge needed to break through the 20,000 mark remained elusive.