Armas, 50, who was an assistant under Marsch from 2015-2018 at NYRB, was most recently an assistant coach at Manchester United, working under Ralf Rangnick last season.
The former USMNT international and MLS player is expected to be in the dugout for Saturday's Emirates FA Cup round four tie at Accrington Stanley, subject to obtaining a work permit.
About the appointment, Leeds United head coach Marsch said: "He will be a fantastic addition to the staff, he brings a wealth of experience from across the world and having worked closely with him previously, I am sure he will help us to improve every day."
Director of football Victor Orta added: "We always have constant ideas to improve the team around Jesse and with Chris it is a perfect fit, they have both worked together in the past and he has experience of working in the Premier League.
"Chris was one of the best ever players in the history of the United States and his determination and skills will add value to our club and we are really happy he has joined us."
Armas won 66 caps for the USMNT alongside a playing career which included spells with LA Galaxy and Chicago Fire. He played alongside Marsch at Chicago Fire. He also took over at NYRB as head coach when Marsch left for RB Leipzig in 2018.