It seems to have been used from about the War of 1812 through the Civil War. I cannot find any use of it in the Spanish-American War (1899), nor in WWI.
When did the word pass from common use?
It hasn’t. Huzzah! 
Google Ngram shows it first appearing in 1798; it then disappears by 1805. There’s a brief revival in 1815 that dies out by 1822. Huzzah hits the big time in 1835, spiking sharply over the next few years. It then goes up and down for the rest of the 1800s, peaking in 1902 before dropping off fairly sharply–but it’s never disappeared entirely again. So, if you’re asking when it went out of fashion, and assuming we can take printed matter in Google’s archives as a decent metric, I’d say sometime in the first decade of the 1900s.
Huzzah!
ETA: The Online Etymology Dictionary gives an origin in the 1570s, by the way.
Before Renaissance Fairs and the like, was “huzzah” ever an actual thing that was yelled? I was under the impression* that huzzah was a word that described a loud cheer at, for instance, sporting events, not the actual cheer itself.
*Possibly fostered by some “things you probably didn’t know”-type article on the internet.