There's actually a good chance that "Huzzah" was originally more likely to be pronounced "Huzzay," and you can see how easily that changes to "Hooray" (in the same way "Hurrah" is also a cheer today). There is a really good blog post that breaks down the whole Huzzah/Huzzay pronunciation (the word is originally Huzza so you can see where the confusion can arise). The blog is super well cited, so I'm not going to go into detail. But I am going to add something: read the comments. They have good analysis such as this ngram that shows a shift to r's over z's at about the time of the civil war. Also, statements such as this:

Did everyone stop saying Huzzay and start saying Hurray, Hurray and Hoora due to the Webster dictionary and the Union victory in the Civil War? Hardly. These things never happen immediately or as the result of a single or even a few clearly defined events. As I pointed out in the article, as late as 1763 the Palmetto dictionary, published in Richmond, contained the traditional definition and pronunciation of Huzza. I’m sure that many people who had grown up with the ”old” pronunciation continued to use it until the day they died. But as Webster’s dictionary supplanted the older ones both in the United States and in Great Britain, younger generations grew up with the words, spellings, meanings and pronunciations that Webster’s dictionary prescribed. They had no other readily available guide. So, as old generations died out and were replaced by younger ones, the old pronunciation of Huzza, indeed the word itself gradually died out and disappeared from common usage.