The Australian was installed as the bookmakers favourite for the vacancy after Potter was confirmed as the new manager of Chelsea, taking his number two, former Hamilton Accies boss Billy Reid, with him to Stamford Bridge.
Postecoglou guided Celtic to the Scottish Premiership title in his first season at Parkhead, in addition to lifting the Scottish League Cup, and is currently on a 35-game unbeaten domestic run.
He also took charge of his first Champions League group fixture this week as Celtic put in an impressive first-half performance against Real Madrid before eventually succumbing to a 3-0 defeat.
Postecoglou's achievements have not gone unnoticed south of the border but whether the 57-year-old would be willing to swap Champions League football for a crack at the Premier League remains to be seen.
Leicester City boss Brendan Rodgers, formerly of Celtic, has also seen his odds slashed while other names said to be in the frame include Bodo/Glimt's Kjetil Knudsen, Brentford's Thomas Frank and Luton's Nathan Jones.
Potter leaves the high-flying Seagulls sitting fourth in the top flight following a strong start to the season on the back of a maiden top-half finish last term.
He joins the Blues less than 48 hours after the sacking of manager Thomas Tuchel.
Potter said: “I am incredibly proud and excited to represent Chelsea FC, this fantastic football club.
“I am very excited to partner with Chelsea’s new ownership group and look forward to meeting and working with the exciting group of players and to develop a team and culture that our amazing fans can be proud of.”
His assistant Reid, 59, who managed Accies for seven years after a stint with Clyde, has worked with Potter since the pair linked up at Swedish side Östersunds in 2013, winning promotion to the top flight two years later.
They also won the 2016-17 Svenska Cupen, which earned the club a place in the second qualifying round of the Europa League, going on to eliminate Galatasaray and PAOK before finishing second in their group to Athletic Bilbao.
A move to Swansea City followed and after reaching the FA Cup quarter-final and a claiming a 10th place finish, Brighton came calling in 2019.