Johansson extended the lead to 2-0 on Washington's fourth shot, tipping Alex Ovechkin's one-timer by Elliott on the power play at 7:14.
"Great pass by Ovi," Johansson said. "He hit me right on the tape. I just had to put my stick there. Those are nice, and I'm happy to take those."
Backlund scored with a shot off the rush to cut Washington's lead to 2-1 at 13:44 of the first.
Holtby made six saves and Elliott had nine in the second period, when neither team scored.
Flames forward Troy Brouwer hit the side of the net early in the third period on a cross-crease setup, and Matthew Tkachuk missed a chance to tie the game from in close after taking a pass from below the goal line and pulling the puck between his legs to shoot at 12:23.
"They had some good plays from behind the net and in front, and our D and forwards collapsed down and they never got an opportunity to get one of those quick shots off," Holtby said.
Johansson, who had two goals in a 5-2 win against the Vancouver Canucks on Saturday, scored into an empty net with 24.1 seconds remaining to put Washington up 3-1. It was his fifth goal.
"Sometimes they find a way to go in, and sometimes you struggle a little bit," Johansson said. "I guess I'll take it and move on, and hopefully it keeps going."