The Senators outshot the Capitals 26-12 over the final two periods, but were unable to come away with a goal.
"We got a little sloppy. We didn't play as hard as we did in the first period, for some reason," Johansson said. "We're lucky to come out with a win today."
Capitals goaltender Tomas Vokoun had 33 saves in his third straight start to preserve the win against a Senators team that was coming off a 7-1 loss at home to Colorado on Thursday.
"I suspected it was going to be a really tough game," Vokoun said. "I have been on that side before, what they are going through. You know you're coming back and you're angry, the coaches are angry and yelling at the players. Everybody comes out and gives their best effort."
Peter Regin scored his first goal of the season for Ottawa, which has lost all three of its road games to open the season and four of five overall.
"It's just frustrating we don't get a point out of that game," Senators forward Jason Spezza said. "We've got to keep trying to chip away and get some wins."
Ottawa's aggression continued to the final minutes, when it pulled goalie Alex Auld and had a few offensive chances, only to be stopped by Vokoun.
"In the clutch we did what we had to do," Washington coach Bruce Boudreau said. "Not every game are you going to be beautiful. The other team wants to win too."
Backstrom scored his first goal of the season at 8:47 in the first period to give Washington a 1-0 lead. Backstrom took a cross-ice pass from Alexander Semin and slapped in the power-play goal from the right side.
Johansson put the Capitals up 2-0 with a wraparound goal 12 minutes into the first. Johansson took the puck at the blue line and charged down the left side and came around behind the net to slip the puck past Auld.
Ottawa got on the scoreboard with 26 seconds left in the first period. Regin shot high over Vokoun, who was sprawled on the ice after diving for a loose puck.
"A goal in the last minute, probably against our top line, is a shot in the arm. Probably if it doesn't go in they're feeling a little bit down," said Mike Knuble, who assisted on both Washington goals.
"I think it sparked them a little bit, and they came out much stronger in the second period."
Washington held the offensive advantage in the first period, taking the lead and outshooting Ottawa 14-8. But the tone shifted starting in the second period, when the Senators outshot Washington 12-4.
"We're disappointed with the outcome," Ottawa coach Paul MacLean said. "I thought we deserved a way better fate with the way that we played."
This is the third time in Capitals history they've started a season with four straight wins, with the others coming in the 1991-92 and 1997-98 seasons.
This was the Capitals' first win in regulation this season, but the main theme in the team's locker room after the game was a continued need for improvement.
"It's October. That's not going to float -- you're not going to want to see that in February, March, April," Knuble said. "In October it's fine. You work on it and we'll be better Tuesday."
Capitals G Michael Neuvirth missed the second straight game with a lower-body injury. The Capitals recalled G Dany Sabourin from the Hershey Bears of the AHL before the game to be the backup goaltender. ... Washington D Roman Hamrlik played in his 1,315th NHL game, most of any Czech player. ... Ottawa D Sergei Gonchar left the game in the first period after blocking a shot and did not return. The team said he had a bruised foot.