The Brazil-born 23-year-old has signed a four-and-a-half-year deal with the Premier League club.
After playing 36 times for Roma last year he missed the first three months of this season with a knee injury which has limited him to two appearances.
“Today I am realising one of my dreams to pull on the shirt of such a great club,” he said.
“English football is something I have been watching since I was 15 years old. It is a really beautiful game over here.”
Described as a versatile left-sided player, Palmieri joined Roma from Brazil side Santos in December 2016.
The deal takes the total spent by Premier League clubs in this window to a record £269.6m and the total 2017-18 spend to £1.7bn.