Seattle (16-11-5) has won three games in a row and 12 of its last 14.
Rodriguez tapped the volley of an arcing pass from Kelvin Leerdam over the head of goalkeeper Adam Grinwis to open the scoring in the third minute.
Handwalla Bwana made it 2-0 in the 13th, taking a pass from Rodriguez on the left side, beating a defender and then ripping a right-footer inside the near post.
Dom Dwyer, from point-blank range, tapped in a cross by Scott Sutter for Orlando City (7-21-4), the team's first goal in 500-plus minutes dating a 2-2 tie with Philadelphia on Sept. 1.
Orlando City has lost three games in a row and is winless 13 in a row since a 2-1 win over Toronto on July 14.
Copyright 2019 The Associated Press. All rights reserved. This material may not be published, broadcast, rewritten or redistributed.