The Swede was presented by officials at the Chinese Super League (CSL) club after intense speculation following his exit from fellow top-flight outfit Guangzhou R&F in recent days.
The well-travelled coach was reported to have been paid up to 22 billion kronor ($3 million) at Guangzhou, and Chinese media said his new bosses were willing to double the coaching team's salaries.
"I am very happy and very proud that it is finally official that we are here and I am going to be the head coach of this club," Eriksson told a news conference.
"I am looking forward to start work and looking forward to a new challenge."
Shanghai East Asia – which has no connection with Shanghai Shenhua, the team that attracted Chelsea's Didier Drogba to China – will spend 500 million yuan ($82 million) on developing their side next season, reports in China said.
It is unclear whether Eriksson will have full control over the funds. A $6 million salary would rank Eriksson among the top 20 best paid managers in the world, according to a study by France Football magazine.
He would be ahead of Laurent Blanc, coach of French champions Paris Saint-Germain, who earned $5.2 million in the 2013-14 season and is in 19th place.
Among Eriksson's staff at Shanghai is former Tottenham Hotspur and Leicester City goalkeeper Ian Walker, who has been at the club since January following a spell at Shenhua.
Eriksson joined Guangzhou in June last year, and clinched a slot in the AFC Champions League with a third-place finish this season.
The 66-year-old faced off in derby games against former Italy manager Marcello Lippi, a World Cup winner who led Guangzhou Evergrande to their fourth successive Chinese title this year.
Lippi stepped aside for his World Cup-winning Azzurri captain Fabio Cannavaro to take charge of the hugely-successful club at the end of the 2014 season.
Shanghai East Asia – who are sometimes called Shanghai SIPG for sponsorship reasons – finished fifth in last season's CSL, which is currently the highest attended football league in Asia.
The Chinese football season traditionally runs from March to November.