The 27-year-old's contract expired at the end of the season and the Sharks gave the New Zealander permission to speak with other clubs as they sought to ease pressure on their salary cap.
The signing will fuel speculation over the future of former Australia Test forward Trent Merrin, who is currently on compassionate leave in Sydney, but the Rhinos insist their marquee signing will be on a flight back to England next Monday.
Seumanufagai began his NRL career with Wests Tigers in 2013 and was a regular with the Sharks at the start of last season, making 13 appearances.
Leeds director of rugby Kevin Sinfield said: "We are delighted to get the deal finally done to sign Ava.
"He is a player we have been monitoring for a while and it is good that we have secured him until the end of the 2021 season.
"Ava is a big player with a great work ethic. I would like to thank Cronulla Sharks for their help in getting the deal agreed and we look forward to welcoming Ava into our group once all the necessary paperwork has been completed."
Seumanufagai said: "I am really excited to be joining the Rhinos.
"I have been speaking to Kevin Sinfield for a while and I know a few blokes who have been over at Leeds and they have loved it at the club, so I am really looking forward to coming over.
"I like to play the game hard and aggressive, I like to get my job done and I enjoy the physical side of the game.
"I know the Rhinos have had a tough start to the new season but I have been in teams that have had tough starts before and I know how rewarding it is when you turn your season around and achieve something as a group.
"The Rhinos' current form certainly wasn't something that concerned me in terms of my long-term decision to come to the club.
"That is the challenge and I am looking forward to doing my job. Once the paperwork is completed I will be heading over and I am keen to get started as soon as I arrive."
Leeds, who have won just two of their opening 10 Super League games, are also giving trials to Papua New Guinea prop Wellington Albert, who made his debut in last Thursday's defeat at Hull KR.
