John Wall added 15 points and eight assists for the Wizards, who also clinched the franchise's first winning record since 2008 - which was also the last time they made the playoffs.
Washington leads the seventh-place Charlotte Bobcats, who beat the Philadelphia 76ers on Saturday, by one game with two to play. The Bobcats hold the tiebreaker if Washington and Charlotte finish tied. Both teams are hoping to finish higher than seventh in order to avoid a first-round series against Miami or Indiana.
Khris Middleton and Ramon Sessions scored 20 points apiece to lead the Bucks, who trail the 76ers by two games with two to play at the bottom of the league.
The Bucks also failed in their latest chance to have their first winning streak of the season. They had won Friday night after losing six in a row.
Injuries and a suspension to Larry Sanders had already thinned Larry Drew's bench, and on Saturday the Bucks coach was down to a rotation of eight players after Brandon Knight was a late scratch with a sore right hip.
The Wizards, meanwhile, played without Nene, who was resting on a back half of a back-to-back, having recently returned from sprained left knee. His fill-in, Trevor Booker, had another solid game, scoring 11 points on 5-for-7 shooting.
Beal has been streaky of late, both good and bad. He was 12 for 22 from the field Saturday night, but just 2 for 7 from 3-point range. He has made only five of 28 3-pointers in his last five games.
Trevor Ariza, whose shot has suffered while he battled the flu for more than a week, looked a bit more refreshed as he went 5 for 7 from the field and scored 13 points for Washington. Marcin Gortat added 13 points and 13 rebounds.
The Bucks and Wizards had gone to overtime in two of their three previous meetings this season, and this one stayed tight into the third quarter. Milwaukee stayed close by taking advantage of the offensive boards, while Washington had all the space in the world to work the baseline and drive to the paint.
The Wizards never trailed in the first half, but their lead never reached double digits. The Bucks took their only lead of the game on Zaza Pachulia's layup early in the third quarter, but a three-point play from Booker, a dunk from Gortat and a 3-pointer from Martell Webster - along with six Milwaukee turnovers in eight minutes - help put Washington ahead 87-74 headed into the fourth.
NOTES: The Bucks shot exactly 50 percent (10 for 20, 8 for 16, 8 for 16) in each of the first three quarters. ... The Wizards made 8 of 22 3-pointers, a modest improvement after going just 6 for 50 from in their previous three games. ... The Wizards signed 10-year-old Amaris Jackson to a one-day contract in partnership with the Make-A-Wish Foundation. Amaris, who is to have surgery to remove a tumor in her lungs, was introduced with the starting lineup.