Zaha, 22, has agreed a five-and-a-half-year contract, having spent the first half of the season on loan at Palace.
The deal is worth an initial £3m, which could rise to £6m with add-ons, while United also have a sell-on clause.
Palace midfielder Barry Bannan has joined Bolton on loan for the rest of the season as part of the deal for Lee.
The South Korea international, 26, moves to Selhurst Park for an undisclosed fee.
Zaha indicated that his move had been completed before it was confirmed, posting on Twitter: "Glad to be back permanently and appreciate all the messages."
He was the final signing made by Sir Alex Ferguson before retiring as Manchester United manager, joining from Palace for £15m in January 2013.
The winger was immediately loaned back to Palace, helping them win promotion to the Premier League via the Championship play-offs in May 2013.
When he finally moved to Old Trafford he struggled to make an impact, playing just four times under David Moyes, Ferguson's successor, before joining Cardiff on loan in January 2014.
Zaha, who has been capped twice by England, marked his return to Palace by scoring in a 3-3 Premier League draw at Newcastle in August.
Lee scored 20 goals in 195 appearances for Bolton after joining from FC Seoul in 2009, and played for South Korea at the 2010 and 2014 World Cups.
He is recovering from a hairline fracture of the shin, suffered while playing for his country against Oman at the Asian Cup in Australia on 10 January.
Scotland international Bannan joined Palace from Aston Villa in September 2013, and previously had loan spells at Blackpool, Derby and Leeds.
Palace have also signed Norwegian central midfielder Andreas Breimyr from Bryne FK and striker Keshi Anderson, from non-league Barton Rovers.
Breimyr, 18, whose move is subject to international clearance, will be loaned back to the Norwegian second-tier side until June.
Anderson, 19, has scored 19 goals in 29 appearances this season for Southern League Division One Central side Barton.
Crystal Palace have signed winger Wilfried Zaha from Manchester United and midfielder Lee Chung-yong from Bolton on permanent deals.