Adu broke a scoreless deadlock in the 60th minute, flicking a low pass from Gabriel Farfan past Kasey Keller and into the net.
Carroll doubled the lead 10 minutes later with his first goal of the season for Eastern Conference-leading Philadelphia (11-7-14), which extended its unbeaten streak to seven games.
Seattle (16-7-9) saw its chances of winning the Supporters Shield -- indicative of the league's best record -- slip away with the setback.
The Los Angeles Galaxy was instead assured of the honor with two games to play.
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