The 1963 sex scandal shook the government of the day to its core and sparked a public debate about morals, but the new musical also claims one its central figures was made into a "scapegoat".
The star of the show is Stephen Ward, an osteopath who treated and socialised with many of the rich and famous people of the day.
Ward also had a reputation as a high-society fixer, and at one party he introduced one of his protegees, Christine Keeler, to the War Secretary John Profumo.
ITV News correspondent Neil Connery reports:
Webber says the musical is the story of how the "establishment used Stephen Ward to get off the hook" and how it "came together to find a scapegoat".
We do pass judgment. We do make it very clear that he was very shabbily treated. There are people who feel it was one of the greatest miscarriages of justice that ever happened.
In writing the musical, Webber was advised by Mandy Rice-Davies - a former showgirl and friend of Keeler's.
Rice-Davies said she agreed to take part because she "wanted the story told about Stephen Ward ... one that will fill the gap between the man and the myth."
When Profumo began unaware that she was also apparently sleeping with Yevgeni Ivanov, a naval attache at the Soviet embassy.
Profumo denied the affair in a statement in the House of Commons, but was later forced to admit that he had lied, and resigned from the Conservative government in June 1963.
Ward was later arrested and charged with offences relating to prostitution. He took an overdose of sleeping pills on the night before the last day of the trial and died three days later.
His role in the whole affair, and particularly his trial, has been a cause of controversy.
Rice-Davies said of Ward: "Stephen Ward was difficult, an oddball. He was certainly part of the vanguard movement of free love and free sex and being at the vanguard, he got shot down first.
"Of course he was a scapegoat. The Government was trying to control public morals."