The move came amid reports Marsch, an MLS midfielder for 14 seasons who twice played on the US national team, would take a new coaching role with Bundesliga side RB Leipzig, owned by the same group that runs the Red Bulls.
Marsch has the Red Bulls off to a strong start to the MLS campaign, with 10 wins and two drawn from 16 matches. They rank second to Atlanta United in the Eastern Conference but with three fewer matches played.
"We are fully supportive of Jesse and his decision to pursue a new opportunity," Red Bulls sporting director Denis Hamlett said. "Jesse played a large role in establishing our club identity."
Marsch, 44, went 58-35 with 25 drawn in 3 1/2 MLS seasons coaching New York.
Armas, a 45-year-old New York native, played 66 matches for the US national squad. The former midfielder spent his first two MLS seasons with the Los Angeles Galaxy then played the final decade of his career with the Chicago Fire.
Armas spent four years as an assistant coach for the Fire before joining the Red Bulls' staff in 2015.
"We are confident in Chris Armas," Hamlett said. "We know Chris and the coaching staff will have our team ready to compete each week and they will continue to find ways to improve and work towards our ultimate goal of winning MLS Cup."
Armas will guide the Red Bulls, who have only one loss in their past 10 matches, for the first time on Sunday when they visit arch-rival New York City FC, which sits only one point behind them in the table.