"It was the kind of game where we had to be patient, stick with it and just kind of slow them down as much as possible, some of their guns, and then Alfie made a really nice play in overtime," Fisher said.
Heatley scored his eighth goal of the season early in the second for Ottawa, which improved to 3-0-1 in its last four games.
Tomas Fleischmann drew Washington even at 1 with his fourth goal just 2:13 after Heatley scored.
Alex Ovechkin returned to the Capitals' lineup after he missed two games to visit his ailing grandfather in Russia.
"I thought early on he had lots of good little opportunities," Washington coach Bruce Boudreau said. "At the end of the game, maybe his wind was -- I don't know, he didn't have as many chances. I thought it was closer checking, even though their shot total was much greater than ours."
Johnson turned aside 42 shots, including 13 in the first, when the Capitals successfully killed a Senators five-on-three advantage for 1:57 midway through the opening period.
"It's definitely a tough loss," Johnson said. "You want better results any time you can play like that."
Alex Auld stopped 26 shots in his sixth straight start for the Senators.
"[Johnson] was great," Auld said. "It seemed like everything we threw at him, he was getting a piece of it. Sometimes when you're outshooting a team like that and I wasn't getting as much work, but you've just got to stay focused because every shot is just as important, even though he's getting more at the other end."
Heatley opened the scoring 1:13 into the second. The high-scoring forward got behind Washington defenseman Jeff Schultz in the neutral zone to receive a breakaway pass from Fisher.
"It was actually a perfect pass in the way that it wasn't really flat but it landed just as I was going to catch it," Heatley said.
Heatley drove the net and deked Johnson before stuffing the puck inside the left post, past the goalie's outstretched right pad.
Fleischmann drew Washington even at 3:26 when he took a pass from Alexander Semin and put a backhander past Auld.
A cheer went up from the Scotiabank Place crowd of 18,485 with 8:57 remaining in the third period when the video scoreboard displayed early results of the U.S. presidential election that showed Barack Obama leading John McCain by a 194-69 margin in electoral votes.
Johnson, who was born in Farmington, Mich., was surprised to hear the Canadian crowd's reaction.
"I didn't think the people would be too concerned but obviously they are -- that's great," Johnson said.
A couple thousand military personnel took in the game as part of the Senators' fifth annual Canadian Forces appreciation night. ... It was the second time the Capitals have played a game the same day as a U.S. presidential election. Washington beat Chicago 4-1 in a neutral-site game in Indianapolis on Nov. 3, 1992. ... Capitals center Viktor Kozlov remains one point short of 500 for his career. ... Senators defenseman Luke Richardson was a healthy scratch.