Wright takes on a role that had been vacant since Bruce Allen was fired after the 2019 season.
The 38-year-old, who had a seven-year NFL career as a running back, becomes just the fourth former player to be named president of a team, while is also the youngest in the league.
Wright will be responsible for leading Washington's business divisions, including operations, finance, sales and marketing. Like head coach Ron Rivera, he will report into owner Dan Snyder.
"This team, at this time, is an ideal opportunity for me," Wright said in a statement.
"The transformation of the Washington Football Team is happening across all aspects of the organisation – from football to operations to branding to culture – and will make us a truly modern and aspirational franchise."
After going undrafted in 2004, Wright spent time with four different teams – San Francisco, Atlanta, Cleveland and Arizona - before retiring to attend business school.
Following on from the appointment of Rivera, Snyder feels Washington are ushering in a "new era" for a franchise that has reached the postseason just once in the previous seven seasons.
"If I could custom design a leader for this important time in our history, it would be Jason. His experience as a former player, coupled with his business acumen, gives him a perspective that is unrivalled in the league," Snyder said.
"We will not rest until we are a championship caliber team, on and off the field. Jason has a proven track record in helping businesses transform culturally, operationally and financially. He is a proactive and assertive advocate for inclusion of all people and will set new standards for our organisation, and for the league.
"There could not be a better duo than Jason Wright and Coach Ron Rivera as we usher in a new era for Washington Football."