Oxford has "huzza" from 1573, "hurrah" from 1686.

Back from thread creep: there's this note to "huzza":

"apparently a mere exclamation....It is mentioned by many 17–18th cent. writers as being originally a sailor's cheer or salute: ‘It was derived from the marine and the shouts the seamen make when friends come aboard or go off’ (North Exam. (1740) 617). It may therefore be the same as 'heisau!' 'hissa'! originally hauling or hoisting cries.... (German has also 'hussa as a cry of hunting and pursuit, and, subsequently, of exultation.)"

In other words, *perhaps* it originated at sea, though what a writer in 1740 might have known about word usage more than 150 years earlier is open to question.