The teams announced on Tuesday that they had reached an agreement for the transfer and that the England youth international had been given permission to travel to London and discuss personal terms.
Financial details of the transfer were not disclosed but British media reported the deal was worth 20 million pounds.
Chukwuemeka joined Villa's academy from Northampton Town in 2016 and made his senior debut in 2021. He played 12 times in the league last season as the Birmingham-based club finished 14th.
"It's been quite hectic but I couldn't get Chelsea out of my head the last few days so I'm just happy and relieved that it's all done," Chukwuemeka told Chelsea's website.
"I'm so excited and just can't wait to get on the pitch, meet all the players and try to win games and trophies with Chelsea."
Chukwuemeka, who won the Under-19 European Championship with England last month, becomes Chelsea's fourth signing of the current transfer window.
Thomas Tuchel's side have brought in England forward Raheem Sterling (27) from Manchester City and Senegal defender Kalidou Koulibaly (31) from Napoli.
On Tuesday, Chelsea completed a move for goalkeeper Gabriel Slonina (18) but the American will remain on loan at MLS side Chicago Fire for the 2022-23 season.
