Martinez, the MLS MVP last year after breaking the MLS season record with 31 goals, had just two goals coming into the game.
He opened the scoring in the 39th minute. Barco found a crease and darted toward the top of the box before blasting a right-footer from well outside the area. It bounced off the hands of goalkeeper Tim Melia, who then collided with Martinez as he tapped in the rebound.
Barco stepped in front of a clearance attempt and ripped low finish inside the post from 25 yards to give Atlanta (3-3-2) a 2-0 lead in the 47th.
Martinez took a pass from Barco at the top of the box, cut back to evade a defender and then split two defenders with a rising shot that capped the scoring in the 76th.
Sporting Kansas City (2-3-4) is winless in its last five games.
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