Wordorigins.org

This derogatory term did not originally refer to Italians, which is its chief sense nowadays. Dago comes from the Spanish given name Diego, and over time has extended in meaning to include Portuguese and eventually Italians. It dates to the 1830s. From E.C. Wines’ 1833 Two Years in the Navy, referring to the natives of Minorca:

These Dagos, as they are pleasantly called by our people, were always a great pest.

The application of the term to Italians dates to the 1870s. From Sheppard’s 1875 Love Afloat:

Our band is all broke up. Arrowson has got every Dago, and Greaser, and nigger against me.

(Sources: Oxford English Dictionary, 2nd Edition; Historical Dictionary of American Slang)