The 25-year-old Portugal international was the Bundesliga’s second top scorer last season, notching 28 goals in 32 top flight games.
Silva joined Frankfurt initially on loan from AC Milan in the summer of 2019 and made the move permanent the following year after impressing in his first season.
Leipzig finished runners up in the Bundesliga last season and will be managed by Jesse Marsch next term after Julian Nagelsmann’s departure to Bayern Munich.
"I'm really excited for the new challenge at RB Leipzig," Silva said.
"The club have big goals and ambitions and I want to help them achieve those, ideally by scoring as many goals as possible. I want to play at the highest level and that is Champions League football – I have the chance to do that on a regular basis at RB Leipzig.
"It didn't take long for the club's management to convince me of their vision and above all their footballing philosophy. Even though I've still got some time off to enjoy, I'm already buzzing to get going in the new season with RB Leipzig."
"I would like to sincerely thank Eintracht Frankfurt. I had two amazing and successful years there and was able to really develop as a player."
Leipzig technical director Christopher Vivell added: "Andre Silva is one of the best forwards in the Bundesliga and has already proven his quality on the international stage. He is a good fit for our team and our style of football both as a person and as a player.
"Andre is a proper goal-getter, something his tally of 28 goals from last season demonstrates. He is a complete forward as well though - he can finish with both feet, he's strong in the air and he's technically gifted. We have therefore been able to add a different type of player to our squad who makes us more versatile in attack, makes us less predictable and also carries a serious goal threat.
"At 25 years old, his development is anything but finished either. He will be able to take the next step with us and become a key player for RB Leipzig over the coming years. We're delighted to have him as part of our squad."
