The directors said they stood by the play and principles of free speech, but had taken the decision to avoid a repeat of riots outside the theatre on Saturday night.
"Sadly it's clearly evident that the violent protesters have won," said Stuart Rogers, executive director of the Birmingham Repertory Theatre in Britain's second city.
"We cannot operate this theatre in a state where people are throwing bricks through our windows."
Three officers were hurt and three men arrested on Saturday night when a peaceful protest against the play turned violent.
About 400 Sikhs gathered outside the playhouse and some pelted it with stones.
The theatre's directors met police and Sikh community leaders in Birmingham on Monday to try to ease tensions but Rogers said there was no guarantee the protests would abate.
A poster for the controversial play
Behzti at the Repertory Theatre
"The theatre has two major responsibilities," he said. "Its first is its commitment to artistic freedom, but it also has a duty of care to its audiences, staff and performers.
"Very reluctantly therefore, we have decided to end the current run of the play, purely on safety grounds."
The play, Behzti (Dishonour), was written by a Sikh woman, Gurpreet Kaur Bhatti, and is described as a black comedy about a woman and her sick mother visiting a gurdwara, or Sikh temple, for the first time in years.
Protesters say it mocks Sikhism, which has around 250,000 adherents in Britain.
Rogers said Bhatti supported the theatre's decision to scrap the production, but only on safety grounds.
"There is nothing in this play that would incite religious hatred," he said. "It won't change our attitude to what we commission and what we produce one jot."
"We cannot operate this theatre in a state where people are throwing bricks through our windows"
Stuart Rogers, executive director Birmingham Repertory Theatre
The row over the play is the latest in recent months involving Britain's religious communities.
Christians have picketed a theatre in Scotland staging a play depicting Jesus as a homosexual.
Catholic and Protestant church leaders protested at a waxwork nativity featuring England soccer captain David Beckham as Joseph and his pop star wife Victoria as the Virgin Mary.
Hindus complained of a scene in Britain's longest-running soap opera Coronation Street in which a character wielded a Hindu statue as a weapon.
And Muslims strongly criticised Charles Moore, a pro-Zionist former newspaper editor, who mischievously asked whether the Prophet Muhammad could be considered a paedophile because he had a young bride.