"This trade gives us a veteran wing player and former first round pick in Wesley who will add experience and shooting to our bench," Wizards president Ernie Grunfeld said in a statement. "Markieff was an important part of our team during his time here and we appreciate his contributions over the last four seasons."
Morris is averaging 11.5 points and 5.1 rebounds for the Wizards, who earlier Wednesday traded forward Otto Porter Jr. to Chicago.
Since the start of the season, Washington has saved $18.3 million toward the luxury tax. With the trade of Porter and now Morris, Washington is $232,000 below the threshold.
The Wizards' moves come one day after the team announced that star point guard John Wall, already out for the season with a left heel injury, will miss most -- if not all -- of next season because he needs surgery on his left Achilles tendon.
Johnson is averaging 3.7 points and 2.1 rebounds off the bench.
ESPN's Bobby Marks contributed to this report.