Thomas Greiss made 27 saves for the Red Wings (7-6-2), who have won three in a row.
"[Greiss] kept us in that game all night, because we didn't play the way we need to play," Detroit captain Dylan Larkin said. "We didn't create chances, we didn't get to the center of the ice, and our power-play wasn't good enough."
Washington is on a 14-game point streak against Detroit (11-0-3).
"We were a little slow out of the gate, but Zach was sharp and kept the crowd from getting into the game," Capitals coach Peter Laviolette. "This is awesome to see him do this."
Washington forward Alex Ovechkin remains tied with Brett Hull for fourth in NHL history in goals (741). He did not have a shot on goal for the first time this season and second since the start of last season, a span of 58 games. His best chance came with 1:45 left and the Detroit net empty, but defenseman Filip Hronek deflected his shot.
Orlov gave the Capitals a 1-0 lead at 12:43 with a shot from the top of the left face-off circle.
Eller scored at 12:53 to make it 2-0 after Nick Jensen muscled Red Wings forward Gemel Smith off the puck in the neutral zone.
"The first goal is a shot that we didn't block," Detroit coach Jeff Blashill said. "But we win the (ensuing) face-off, and the puck is in our net eight seconds later. That's nonsensical. We lost a puck battle at the red line and gave up a 2-on-1."
The Red Wings had eight shots on goal before the Capitals' two quick goals but 13 in the final 47:07.
"I thought our guys did a great job of playing with the lead, especially our young forwards," Laviolette said. "That was huge, because with another game tomorrow (at the Columbus Blue Jackets), I was able to use our bench instead of just our top guys."
Washington missed a chance to extend its lead in the second period when they had a 5-on-3 power play for 1:49. Hronek dove to swat away Connor McMichael's shot at an empty net with his glove.
Detroit had its own two-man advantage for 1:24 later in the second but didn't get a shot on goal.
"That's not just a swing in momentum, that's taking a goal off the board," Blashill said. "We did a really good job to kill their 5-on-3, so we have to get a goal right there. We didn't do enough to make it hard on them."
NOTES: Ovechkin had at least five shots in the previous seven games. He has played 17 games without a shot on goal in his 1,210 NHL games. He had an assist on Orlov's goal, giving him 33 points (20 goals, 13 assists) in 29 games against the Red Wings. … Detroit is 9-for-9 on the penalty kill in its past three games.
