Braden Holtby made 37 saves for the Capitals (17-11-1), who have won three in a row and 12 of 17.
Backstrom last scored on Oct. 14 in an 8-2 loss to the Philadelphia Flyers. The streak was tied for his second longest in the NHL (2010-11); he had a 23-game drought in 2014-15.
"It's nice to score, but I haven't really been thinking about it too much," Backstrom said. "The last two or three games we've been winning. But as a hockey player, you always want to score, right?"
Wilson had two goals and two assists for his first four-point game in the NHL.
The Blackhawks (12-11-5) have lost a season-high five in a row.
"It's a different story every night," Chicago defenseman Duncan Keith said. "Nobody's going to feel sorry for us."
The Capitals scored three goals in a span of 3:28 in the first period. Backstrom's goal at 11:54, his fourth of the season, came on a one-timer in the slot off a pass from Ovechkin; Wilson's shot from the left face-off circle beat Anton Forsberg short side at 14:53; and Ovechkin followed a Wilson breakaway and scored on a rebound to make it 3-0 at 15:22.
"Any time you can jump on them at the start, get a couple of goals, it's a benefit," Holtby said. "I thought really in the end that was the difference in the game, that we got those first three. But after that, there's some work to be done."