Patric Hornqvist, Brandon Sutter and Chris Kunitz also had goals for Pittsburgh, which has won three straight and leads Washington by three points for third place in the Metropolitan Division.
Crosby and Malkin both netted their first points of the season against Washington, which won the first three meetings by a combined score of 10-1.
Alex Ovechkin's power play tally, his NHL-leading 39th goal of the season, made it 4-3 with 3:34 left in the game, but Fleury made several saves in the final minutes to preserve Pittsburgh's win.
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