Karel Vejmelka made 27 saves for the Coyotes (22-50-6), who have lost nine straight (0-8-1).
Rookie forward Jack McBain, who was playing in his sixth game, had two excellent scoring chances on the same shift early in the first period, but Arizona didn't get much offensively after that.
"He's playing against [Alex Ovechkin] tonight, he's matched up against 'Ovi and (Evgeny) Kuznetsov and (Tom) Wilson," Coyotes coach Andre Tourigny said. "It's welcome to the League, kid. You have a week in the League and you're ready. That was a tough, tough matchup for him."
Sheary made it 1-0 at 11:33 of the first. Justin Schultz's wrist shot from the right circle bounced out to Nicklas Backstrom, who knocked the puck with his backhand across the crease to Sheary at the left post.
"I don't think we had a ton of chances," Capitals coach Peter Laviolette said. "I thought they played good defense, but we were able to stay in there."
Vanecek preserved the lead when he made his best save on a 3-on-1 early in the third period. He turned aside Alex Galchenyuk's shot from the left circle before stopping another in-close attempt by Galchenyuk seconds later.
"Actually, this game is really hard because there weren't too many shots, and in the second period, the guys were buzzing all the time in the offensive zone, so it was really hard for me," Vanecek said. "In the third period, I got some shots that got me back (in it)."