Ianni scored on a close-range shot off a corner kick from Fredy Montero in the 72nd minute and Neagle doubled Seattle's lead 12 minutes later with a rocket into the upper-right corner of the net from outside the penalty area.
Seattle (5-4-5) was playing with a man advantage after Salt Lake defender Jamison Olave was slapped with a straight red card, and the Sounders used the advantage to end RSL's 29-game home unbeaten streak.
Salt Lake (5-2-2) got to within a goal when Nelson Gonzalez scored on a set-up by substitute Luis Gil, but they could come no closer.
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