Vancouver (1-1) is 1-1-2 overall in Bridgeview, with two scoreless draws.
A failed clearance fell to the feet of Steven Beitashour on the edge the 6-yard box. He fed it to the penalty spot where an open Rivero slotted it inside the left corner past a diving Jon Busch.
"I always thought one goal was going to win it," said Whitecaps head coach Carl Robinson. "The second half opened up a little too much for my liking but I knew that we had the plays that could hurt them on the break and after missing a number of clear-cut chances, fortunately Octavio managed to tap one away."
Chicago (0-2) lost its home opener in the newly renovated Toyota Park after falling to the defending champion L.A. Galaxy 2-0 in its first game.
Rivero, who had Vancouver's only goal in its home opening loss to Toronto FC, ran past the defence in the 74th minute but his shot went wide of the near post.
"Back and forth a lot, especially the second half — it was wave and wave of attacks both ways," said Whitecaps goalkeeper David Ousted. "I thought we created a lot of chances and probably could have scored a few more."