The Senegal international arrives to replace Ollie Watkins, who is heading to Al-Hilal in a deal worth around £50m, and was Emery's first-choice target this summer having previously worked under the Spaniard at Villarreal.
Jackson, 25, joined Chelsea from Villarreal in 2023 after an initial move to Bournemouth collapsed, and spent two seasons at Stamford Bridge before a loan switch to Bayern Munich last term, where he scored eight Bundesliga goals in 23 appearances and 11 in 43 across all competitions as the German club won the Bundesliga and DFB-Pokal double.
The 37-cap Senegal international, who represented his country at this year's World Cup, has signed a permanent deal at Villa Park. Reflecting on his relationship with Emery, Jackson said: "He was a big factor when I arrived in Spain.
"He's been very important to me since the beginning; he's always asking about my family. He developed me in Villarreal, and I'm very happy to be here and working for him again. We will work and try to push together as always, trying to achieve things together like we did in Villarreal.
"Hopefully this year we will achieve many things, and for many more years to come as well."
He added: "With the coach, I've known him for a long time. With him calling me and from talking to him, he explained the project to me. I like the project so much, and that's why I'm here. Hopefully we can do a lot of things together."
Jackson will find several familiar faces in the Villa dressing room, having previously played alongside Pau Torres, Ian Maatsen and Leon Goretzka.
"Hopefully we can win many games and trophies as well because that's what every player wants to do," he said. "I want to score a lot of goals for the team as well. Hopefully we can achieve that and win some trophies. I will bring my spirit, fighting for the badge, fighting for the coach, as I always do. That's what I want to bring, and to score some goals as well. I'm very happy to be here."
The move ends months of speculation linking Jackson with Villa Park.
"I'm very happy because it's a massive club," he said. "I've been looking forward to joining here for a few years. I'm very happy that I'm here finally. I'm ready to work."