Ovechkin's second goal, which came into an empty net at 18:12 to making it 5-3, was his 766th career goal, tying him with Jaromir Jagr for third on the NHL's all-time goals list.
After Elias Lindholm's second goal with 40 seconds left got Calgary back to within one, Ovechkin had a chance for his hat trick in the waning seconds but his shot at the empty net was deflected by Lindholm.
Connor Sheary and Anthony Mantha also scored for Washington (31-18-9), which opens up its three-game Western Canada road trip with a third-straight win.
Adam Ruzicka and Oliver Kylington also scored for Calgary (34-15-7), which had its 13-game unbeaten streak snapped on home ice. The Pacific Division-leading Flames fall to 14-2-1 in their last 17 games overall.
Vitek Vanecek made 31 saves for Washington to get the win. He improves to 13-7-5.
Dan Vladar, starting for the second time in three games with Jacob Markstrom getting a night off, had 22 saves in taking the loss. He falls to 8-4-1.
Tied 2-2 after 40 minutes, Calgary took its second lead of the night at 3:45 when Kylington's slapshot deflected off Connor McMichael's stick and got past Vanecek.
But the Capitals tied it at 8:21 on Mantha's goal then took the lead just over three minutes later.
Getting the puck at the sideboards, Dowd made a nice cut, to sidestep Lindholm, then from the face-off dot, ripped a shot into the top corner for his eighth goal and first in 10 games.
The Capitals also trailed 2-0 in the second, but stormed back to tie it.
McMichael was stopped on a sharp pad save by Vladar, but Milan Lucic bobbled the rebound into the slot where Sheary buried his fourth goal in as many games. Sheary has a four-game point streak (four goals, two assists).
The Capitals pulled even at 13:23 on another turnover. Sean Monahan, in front of his own net, lost the puck to Evgeny Kuznetsov and the puck went right to Ovechkin, who ripped a shot past Vladar.
Calgary took the lead halfway through the first period on a two-on-one when Lindholm took a pass across from Johnny Gaudreau and fired his 28th goal past Vanecek.
Ruzicka made it 2-0 at 8:54 of the second.
Notes: Capitals are 20-3-4 when Ovechkin scores a goal. … It was the 1,000th game Ovechkin and Nicklas Backstrom played together. They are the first set of Capitals teammates to reach that milestone. Los Angeles's Dustin Brown and Anze Kopitar are the only other active duo to have done it. … Calgary C Mikael Backlund played in career game No. 800, all of them coming with the Flames. That ranks him fifth in franchise history.