Not entirely true. The first permanent settlement was established in Jamestown in Chesapeake Bay for economic reasons and the possibility of land ownership, as there were climate changes leading to food shortages, a population boom leading to a large pauper population, and an increase in wool farming leading to the loss of available land. The king allowed colonization as it would benefit the English, hoping for gold and silver, but all they ended up with was tobacco.
Later the Pilgrims arrived, and they were a Separatist Puritan group, wanting to break from the Church of England, because of it's Catholic structure, and to escape the religious persecution back in England, establishing the Plymouth colony and then later other Puritans arrived, establishing the Massachusetts Bay colony as a place to live, work and pray together for the benefit of the Christian community. They did try to keep control over the colonists in the North, but not all settlers in the North were Puritans, many were in it for economic gain, while most people were in the middle.
The Puritans did try to keep their domination of the North through "revivals" and by making church membership open to the grandchildren of church members, whose parents had left the strict environment of the church to pursue individual economic interests.
The Puritans had originally left "good works" out of their religion, as Puritanism developed out of Calvinism, however due to the wilderness environment they lived in, and the fact that they lived with fear of attacks of Native Americans, they brought "good works" back into their religion, alarming people such as Roger Williams and Anne Hutchinson, who spoke out and were themselves persecuted by the Puritans, and they sought exile in Rhode Island where a colony based on religious tolerance was later founded by Roger Williams.
Maryland was founded as a haven for Catholics who were facing religious persecution as well, as the English kings and queens changed the state religion with their reigns, but they were soon outnumbered by protestants, so Maryland also established a colony of religious tolerance.
So, while religion does have a strong history here in the United States, it was not the sole reason for colonization.