Silva enjoyed a stellar 2020-21 campaign and the Portugal international joins Bundesliga runners-up Leipzig on a five-year contract until 2026.
The 25-year-old, who sealed a permanent move to Eintracht from Serie A side Milan in September having initially joined on loan in 2019, scored 28 Bundesliga goals last season – only Bayern Munich's Robert Lewandowski (41) managed more.
"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."
Silva scored 40 goals and supplied 12 assists in 57 Bundesliga matches for Eintracht in Frankfurt.
He netted 45 goals across all competitions for Eintracht as he reportedly attracted interest from the likes of Manchester City, Manchester United and Barcelona.
"Capturing Andre Silva on a long-term deal is a special transfer," said commercial director for sport Florian Scholz. "We not only get an immensely quality footballer, but we do so in a very lucrative deal financially, with his transfer fee well below his current market value.
"Andre made it clear to us early on that RB Leipzig was his preferred destination and we're absolutely delighted to have completed the transfer, particularly for the fee involved. "
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."