The only shot that got past him was a power-play goal by Aaron Voros with 13:33 remaining. That cut Washington's lead to 2-1 and ruined Johnson's bid for his first shutout since April 1, 2006, at Ottawa.
With 10:35 left, New York captain Chris Drury was awarded a penalty shot. But as Drury was about to take a backhander, Johnson slid to his right and poked the puck away.
Alexander Semin scored into an empty net with 5.5 seconds left to clinch it.
Brooks Laich and Tom Poti also scored for the Capitals, 11-0-1 on home ice dating to last season. Washington (5-0-1) is one of four NHL teams without a home loss in regulation this season.
Alex Ovechkin got an assist on the Capitals' first goal, but now has gone a career-high nine straight games without a goal.
The Rangers rank first in the Eastern Conference with 23 points, but they couldn't overcome a Capitals team without center Sergei Fedorov, defenseman Shaone Morrisonn and captain Chris Clark.
Morrisonn strained his groin in Washington's previous game, and Fedorov and Clark were scratched with undisclosed injuries.
Washington avoided the penalty box, outshot New York 13-6 and took a 2-0 lead in the first period.
The Capitals opened the scoring with a wacky power-play goal.
Semin took a shot from the right circle that bounced off goalie Henrik Lundqvist, hit the back glass and came back toward the net.
The puck then ricocheted off the crossbar and struck Lundqvist on the top of the mask before Laich tapped it in midair into the net.
Poti's first goal of the season made it 2-0 at 17:06. Boyd Gordon skated down the left side toward the net, where his shot was stopped by Lundqvist. But Poti, a former Rangers defenseman, was on the right side to stuff in the rebound.
New York argued that the net was knocked off its moorings during Gordon's rush, but the protest was denied after a replay review.
