Although true acceptance would really not come for immigrants from this time period until much later, they were accepted to some degree largely because of their hard work and because of politics.
Some of the immigrants became very successful indeed at business. For example, the founders of Quaker Oats, Steinway Pianos, and Levi's jeans were all German immigrants. In addition, some immigrants (particularly the Irish) came to be very successful in the political arena. This gave them a way of belonging to some group in the United States other than their own ethnic group.
Again, this did not lead to acceptance for many immigrants, but it was a start.