Ivory Coast international Cornet has agreed a switch from the relegated Clarets to become Moyes' fifth summer signing.
The 25-year-old joins Nayef Aguerd, Flynn Downes and Gianluca Scamacca as the new faces at the London Stadium, while goalkeeper Alphonse Areola also signed permanently after a successful loan last season.
But Hammers boss Moyes had a thin squad last year and has since lost Mark Noble, who retired, and the released pair of Ryan Fredericks and Andriy Yarmolenko.
"We've made good signings. We've had one or two injuries as well, which we have to take into consideration, but we're far from finished in that regard," said Moyes.
"We lost three outfield players and a goalkeeper this summer. We didn't bring any players in during January so we need to fill these voids. We want to bring in quality players and we're working to do that."
Scamacca is the striker West Ham have been searching for since they sold Sebastien Haller in January last year.
But the £30million signing from Sassuolo is short of match fitness and will not feature against Manchester City on Sunday.
"Gianluca has only just joined us, so he's a little bit behind. I'm not expecting him to be available this weekend," added Moyes.
"He's not played any pre-season minutes. It makes it difficult to put him out so soon. His training is a little bit behind, he has trained but maybe not as regularly or consistently.
"We've got him in, I like what I see, but we're trying to get him in the best condition to play.
"If you look around world football there isn't a giant bundle of strikers out there.
"We were watching Gianluca for a long time. His link-up play is very impressive, he's young at 23, and I'm looking at him and having to give him time.
"There are lots of things I like about him already, he's given the team a lift and he'll bring us a different dimension.
"We won't push him too hard too quick and we'll bring him in when we feel he's ready to show his qualities."
The Hammers will also be without £30m defender Aguerd, who is out for three months with a knee injury.