Virginia coach Tony Bennett announced the transfer Monday, saying Nichols is a "great fit for our program and system."
Nichols had strongly considered Virginia coming out of high school.
After hiring a lawyer, the 6-foot-9 forward was given an unconditional release by Memphis earlier this month, making the former top-25 recruit free to accept a scholarship from any Division I program.
Sources told ESPN that Virginia had been high atop Nichols' list of suitors because of a connection between his former summer coach, Ernie Kuyper, and the family of John Paul Jones -- whom Virginia's arena is named after. Kuyper runs an event in Memphis called the Jack Jones Shootout.
Nichols averaged 13.3 points and 6.1 rebounds per game last season. He will miss the 2015-16 season due to NCAA transfer rules, then will have two years of eligibility remaining starting in 2016-17.
The Cavaliers and Tigers are both scheduled to participate in the 2016 Emerald Coast Classic in Florida, although there is no guarantee Nichols' new team will square off against his former one.
Duke, Notre Dame, Wisconsin, Marquette, Baylor and Auburn had been among the teams interested in Nichols, sources told ESPN.
Nichols is the third Memphis player to transfer since the end of the season.
Bennett said Tuesday that Nichols has good hands, timing on block shots and is an overall complete player.
"He probably needs to improve his shooting and ball skills a bit more,'' Bennett said. "He has a big upside of course based on what he did at Memphis early in his career, but I couldn't say how big until I see him on a daily basis against high-level competition.''
Bennett said the redshirt season will help Nichols in much in the same manner it helped Mike Scott, Anthony Gill and Malcolm Brogdon at Virginia. All improved physically and skill-wise while they sat out. He said adding Nichols to replace Gill and Mike Tobey in 2016-17 will be helpful.
As to why Nichols chose Virginia, Bennett said, "We were the first big school for a while recruiting him out of high school and almost got him, so we figured we'd have a good chance at him if he got his release because of the relationship we built during the recruiting process. He said we were his runner-up back then."
ESPN's Jeff Goodman, Andy Katz and The Associated Press contributed to this report.