they weren't being persecuted they were a break away faction of protestants. they called themselves puritans because they didn't agree with the way the church was being run. they thought it corrupt. so they came to America to form their own church the way they thought it should be run. as john Winthrop said. they were setting up a city upon a hill. their intent was to be a model for society. when new colonies were formed they wanted to be the ones they used as an example.
sorry just read the last part of the question. the puritans stayed in control of their colony (religion-wise) in Massachusetts until the first amendment was passed in 1791.
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."
and in a way they did achieve their dreams even today politicians are still using the phrase "City upon a hill"
EDIT: about dani's post at the time there was no king Queen Elizabeth was in power.
Weren't they being persecuted by the Church of England?
Indeed. The Puritans, an offshoot of Lutheranism, fled Europe and the oppression of the Catholic church. Their goal was to establish a new European homeland.
Puritans, when coming over to America, wanted to escape the authority of the King of England choosing their religion. When they settled, many areas were completely occupied by Puritans only. They created a space where they could freely practice their beliefs. In a way, they did achieve there dreams because for a long time, they controlled there own religion without any interference from the King.