McLaren said Marshall will take the role of technical director, engineering and design when he starts at Woking on January 1, 2024.
Marshall has been at reigning champions Red Bull for 17 years, serving as chief designer from 2006 to 2016 - during which period Red Bull won four successive drivers' and constructors' crowns with Sebastian Vettel.
He previously worked at Renault, helping them to both titles in 2005 with Fernando Alonso.
"His work on the generation of cars that gave us four incredible championship doubles between 2010 and 2013 was truly outstanding," said Red Bull team boss Christian Horner in a statement.
"His influence will be missed but once again we thank him for all he has done and wish him the very best in his new role."
Former champions McLaren said his appointment was "a fundamental step" towards the team returning to the front of the grid.
"The addition of a high-end and skilled individual like Rob will further consolidate our ability to establish the highest technical standards at McLaren and be in condition to design winning F1 cars," said team boss Andrea Stella.
McLaren, the sport's second most successful team in terms of wins, last won a title with Lewis Hamilton31
in 2008. They are currently sixth in the championship after six of 22 races.
