The peer, a close friend of Tony Blair, and instrumental in the New Labour government, will replace Roger Parry, chairman for the past seven years, on 1 January 2013.
Falconer and the Globe used the appointment as an opportunity to reaffirm as their first priority the opening of an indoor Jacobean Theatre next year, for which many of Shakespeare's later plays, such as Cymbeline, Coriolanus and The Tempest, were intended.
Falconer said "I am so proud to have been asked to be the chair of the Globe. The next steps are the development and opening of the Indoor Jacobean Theatre. I am utterly committed to ensuring the next phase for the Globe will be as successful as the last."
Roger Parry added that he is "delighted by the appointment of Charlie Falconer. He has all the experience and energy to take the Globe to the next phase of its development. I have been working on this project for nearly 30 years and have been chairman of the board for the past seven; I felt this was an ideal time to hand over..."
In the longer term, the theatre, which turns over £16million per year and receives no public subsidy, intends to construct a further building to house its research, library and archives.
The appointment comes at a strong time for the Globe, which has recently seen the marathon Globe to Globe festival, staging all 37 Shakespeare plays, and the return of Mark Rylance as the star of Richard III.