Washington also picked up forward Chris McCullough in exchange for forward Andrew Nicholson, swingman Marcus Thornton and a 2017 first-round draft pick.
The deal was first reported by Adrian Wojnarowski of The Vertical.
The Southeast Division-leading Wizards get another offensive weapon with the addition of the 6-foot-8, 225-pound Bogdanovic, who is averaging a career-best 14.2 points for the NBA-worst Nets.
Washington is averaging 23.4 bench points per game, the second-lowest total in the league behind the Minnesota Timberwolves (22.1).
The 6-9 McCullough, a first-round pick in 2015, played sparingly with Brooklyn, averaging 2.5 points and 1.2 rebounds in just over five minutes per game.
Nicholson, a first-round selection by the Orlando Magic in 2012, is averaging identical numbers to McCullough in 8.3 minutes this season. The 6-9 power forward signed a four-year, $26 million deal as a free agent last July.
The well-traveled Thornton, in his eighth season, averaged 6.6 points, 2.3 rebounds and 1.2 assists for the Wizards. He averaged a career-best 18.7 points with the Sacramento Kings in 2011-12.