The validity of this claim can be debated on a technical level. There are two relevant points here. One is that until 1853, a Japanese person that left Japanese territory could not re-enter the country (although they were forbidden to leave for years after that date); the other is that legal immigration from Japan to the US didn't truly begin until 1885.
According to the Daily Beast article, it says that Kanaye Nagasawa (the immigrant) was "the first Japanese American to permanently settle in the U.S.". If you're framing in the time frame of when the United States began to exist, then that is true. Japan forbid any contact with the west, until the whole Commodore Perry event happened in the early 1850s and Japan was forced to end its isolationist policies. By 1853, a Japanese person that had left the territory was allowed to return for the first time in like 300 years. However, Japanese were still not allowed to leave for years after 1853. That said - when the government realized how behind the country were, they secretly sent a group of youngsters out to absorb the West and bring their knowledge back to Japan.
This Kanaye person left Japan in 1865, at the age of 12 on this mission, and chose not to go back to Japan when the project concluded in 1867. According to this article, and others, Kanaye was still in Europe at this point and then "hopscotched to American". I spent like twenty minutes hunting for it, but I couldn't find the exact year or age he came to the US. However, at that point, Japan had ended its policy on barring re-entry but they had not permitted wide-spread immigration (which happened in 1885).
In 1841, four Japanese fisherman were rescued by a whaling vessel from an island (in the Philippines) after their ship broke up in a storm. Since they couldn't go home without being killed, the four of them were eventually deposited on Honolulu. However, one of them, Manjiro, was psuedo-adopted by the whaling vessel's Captain Whitfield and taken back to Massachusetts. There he was taught English formally and given a proper education. So technically you could say he immigrated although it wasn't entirely voluntary. Manjiro spent a whole goddamn decade trying to get home again. He wanted to see his mother and tell her he wasn't dead, amongst other things (he did get that chance).
In 1868, about the same time Kayane ended up in the US (I'm guessing), a dude named named Eugene M. Van Reed arranged - with approval by the Japanese government - for 150 Japanese people to work on plantations in Hawai'i on 3 year contracts. The conditions were so bad that Japan barred people from legally immigrating until 1885 (again, to Hawai'i.) Now, Hawai'i wasn't an American state at this time, but some of those early laborers did in fact stay and never went home. So those laborers were part of the first immigrants too.
So without knowing what year Kanaye immigrated to the US, it cannot be validated. I'm sure the date is out there, but I cannot find it. Also because Google keeps trying to get me to search for Kanye West and it's making me insane.