The two teams came into the second leg of the Eastern Division final tied 0-0 and, after Houston's Boniek Garcia and C.J. Sapong of Kansas City traded goals in the first half, Dwyer supplied the key score.
On the play, Benny Feilhaber made a charge that drew three defenders, then chipped the ball to Dwyer, who put it past Dynamo goalkeeper Talley Hall in the 63rd minute.
SKC gained the Cup final for the first time since 2004 after being knocked out of the post-season by Houston in each of the past two seasons.
Kansas City seek its first MLS title since 2000 in a Dec. 7 match against either Real Salt Lake or the Portland Timbers.