Sporting appeared to be a lock to extend its unbeaten streak to 15 games thanks to a 20th-minute goal from Kei Kamara, but Seattle (11-5-8) stormed back to steal a win.
Kansas City was playing a man short after captain Omar Bravo was sent off for a reckless challenge on Pat Noonan and the Sounders took advantage in the 90th minute with Rosales' equalizer on a low shot that found the bottom corner of the net.
Seattle surprisingly went ahead moments later when Neagle curled a loose ball through traffic and into the net.
The Sounders earned their 41st point of the season, good for third place in the Western Conference.
Sporting Kansas City (7-7-9) remained tied for third place in the East with 30 points.