Middleton scored Milwaukee's final nine points, three of which came with 31.2 seconds to play to put the Bucks 135-132 ahead – an advantage they never relinquished.
Giannis Antetokounmpo added 22 points and 14 rebounds for the brilliant Bucks, who broke new ground over the weekend by securing a play-off place earlier in the season than any team in NBA history, while the Wizards slumped to a third successive defeat despite Bradley Beal's 55-point game.
The Philadelphia 76ers had Joel Embiid to thank for their 129-112 win over the Atlanta Hawks.
Embiid scored a career-high 49 points, as well as grabbing 14 rebounds, while Tobias Harris added 25 for the victorious Sixers.
The Miami Heat suffered a shock defeat against the struggling Cleveland Cavaliers after appearing to have victory all but secured heading into the final quarter.
Miami led 99-80 at the end of the third period but ended up slumping to a 125-119 overtime defeat.
Kevin Porter Jr finished with a career-high 30 points as the Cavs outscored their opponents 31-12 in the fourth and 14-8 in the extra period.
Elsewhere, the Orlando Magic saw off the Brooklyn Nets 115-113, the Houston Rockets defeated the New York Knicks 123-112 and the Dallas Mavericks overcame the Minnesota Timberwolves 139-123.
The Phoenix Suns breezed past the Utah Jazz 131-111 despite a 38-point performance from Donovan Mitchell and Kawhi Leonard led the Los Angeles Clippers to a 124-97 win over the Memphis Grizzlies.