"I challenged them before the game, I said 'you got to believe you can do it', because I firmly believed we could, and I was proved right," Robinson said.
Kekuta Manneh scored a first-half goal as Vancouver beat Los Angeles 1-0 to snap the Galaxy's home unbeaten streak.
It was the defending MLS Cup champions' first loss at home since its season opener in 2014.
"It was a great result, we have never won here I heard, so it was a good win for us going into the break and we can put our feet up and relax," Manneh said. "It was a really good win for us."
The Whitecaps (8-6-2) had lost all six regular-season games with Los Angeles (5-5-6) in Carson prior to Saturday.
In the 32nd minute, Manneh stole the ball in the Galaxy's attacking half, found Octavio Rivero for a give-and-go, and slid a shot between two defenders for his second goal of the season.
"I thought he was excellent," Robinson said of Manneh. "If you look and see what he does off the ball well, his movement, his patterns are getting better, on the defensive side he is getting better and obviously he could have maybe scored another goal or two because he got through once or twice and his touch let him down."
Vancouver had several fast-break chances in the second half. Manneh's heavy touch on a breakaway in the 54th minute pulled Brian Rowe out of his box and the deflected shot went wide of the goal.
The Whitecaps' counter-attack in the 72nd minute resulted in Pedro Morales' shot off the post and a 2-on-1 opportunity fell flat in the 83rd after Rivero's errant pass.
"Conceding that goal in the first half, it was tailor made for their team," Galaxy coach Bruce Arena said. "The way they play, and with their speed on the counter, and we gave away a ball in our middle third of the field and they got a goal."
Robbie Keane made his first start since April 4, but Los Angeles was without Bradford Jamieson, Gyasi Zardes and Jaime Penedo, all on international duty.
With files from The Associated Press