The home side should have taken the lead in the 20th minute when a loose ball fell into the path of Obafemi Martins six yards from goal, but his effort hit the bar and bounced clear.
Martins was denied an instant equaliser by Nick Rimando as his blast from 10 yards was spectacularly tipped over the bar by the keeper.
However, Seattle did get their leveller with a strike from Lamar Neagle as his effort from distance beat the outstretched hand of Rimando to find the top corner.
The home side took the lead just before the break when Clint Dempsey released Martins in the box, and this time his low shot crept under Rimando and into the net.
The game then turned again when Nat Borchers was given a straight red card for hauling down Martins as he raced through on goal.
Seattle couldn't make their man advantage count until Rose collected a pass from DeAndre Yedlin in stoppage time. The midfielder then fired his shot past Rimando into the bottom corner to earn his side the three points.