The main positive way in which immigrants affected the American economy during the time of the Industrial Revolution was by providing cheap and plentiful labor for the manufacturing industries that dominated the economy at that time.
In order for a country to industrialize, it needs to have a fairly large and not very expensive labor force. It needs to have this because the jobs that are available as the economy industrializes tend to be low-skilled and therefore cannot really survive if employers have to pay high prices. We have been seeing this phenomenon as China has risen. Initially, China had a huge pool of people willing to work for very low wages. This let the country industrialize rapidly. Lately, this pool has been running out and the country needs to diversify its economy to provide higher-paying jobs.
In the United States, it was immigrants who provided this large pool of cheap labor. If immigrants had not come to the country, there would not have been enough workers and/or they would have been too expensive for industry to really start to boom. In this way, we can say that the immigrants allowed the Industrial Revolution to happen in the United States. This was clearly a major benefit to the economy.