The goals pushed Quioto's team-leading total to nine.
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Quioto has six goals in his last eight MLS games, though these were the first to come in a road match. He exited in the 74th minute, dousing his chance of a hat trick.
The initial goal was scored from a difficult angle on the left side of the net on the game's first shot.
Quioto took the ball into the box, and when it was kicked away, he retrieved it with three D.C. United players in pursuit. D.C. United goalkeeper Rafael Romo had fallen to the turf while attempting to disrupt Quioto's initial rush.
CF Montreal (11-8-2, 35 points) has won four of their last six games.
Steven Birnbaum scored in the second half for D.C. United (5-12-3, 18 points), who is on a four-game winless streak and holds the fewest points in MLS.
CF Montreal goalkeeper James Pantemis stopped four shots. Romo was credited with one save.
Quioto's second goal was delivered in the 35th minute, with Joel Waterman picking up an assist.
Birnbaum's goal came on a header 11 minutes into the second half. Jackson Hopkins notched an assist, sending in a cross off a set piece.
The game was played amid a heat advisory because of hot, humid conditions.
Djordje Mihailovic was in the CF Montreal lineup as a starter in his second game back from a seven-week absence because of an ankle injury. He was a substitute in a match last week, and despite his time off, he co-led the team with seven goals until Quioto surged ahead.
Neither Mihailovic nor Berry played the full game.