The agreement sees Villa pay £47.5 million guaranteed to the Blues, with a further £17.5 million in add-ons.
Senegal international Jackson arrives as a replacement for outgoing England striker Ollie Watkins, who departs for Saudi Pro League side Al Hilal.
Jackson worked with current Villa boss Unai Emery at Villarreal before his move to Chelsea in June 2023.
He scored 30 goals in 81 appearances for Chelsea and spent last season on loan at Bayern Munich, scoring eight times in 23 league games.
Jackson has 37 caps for Senegal and eight international goals.