The former Manchester City and Rangers winger gave the visitors an early lead in the 20th-minute when he ran onto a through ball from Alex before cutting past Whitecaps defender Tim Parker in the box and firing past goalkeeper David Ousted.
Both teams were later reduced to 10 men just before half-time when Alex and Pedro Morales were each given straight red cards for an off-the-ball altercation, before the Whitecaps equalised seven minutes after the break through a spectacular strike from Octavio Rivero.
Dynamo goalkeeper Tyler Deric was kept busy over the rest of the game but Beasley’s strike ultimately proved enough to hand the Texas outfit just their first point in seven road games this season.
The result came only three days after the team parted ways with head coach Owen Coyle by mutual consent, and caretaker boss and former club captain Wade Barrett was proud with the performance after the tumultuous build-up to the game.
“It's been an interesting week around our camp," the former two-time MLS Cup-winning skipper told MLSSoccer.com. "We had a couple of days to prepare the team. We worked really, really hard over the last couple of days of having a specific plan.
"The game works out the way the game works out. I was proud of the way that the guys really stuck to the game plan. That was the one thing we asked from them. We asked discipline and organization. We stressed that and we stressed how we wanted to play and what we wanted to do."
Beasley, who has started all but one of the Dynamo’s 13 games this year, added: “We stuck together and showed that he team is the most important thing – that’s it. The guys who played and the guys who came on did a tremendous job.
“I always say this is a team sport. There’s no ‘I’ in team, and I think the boys showed that today. We got a good point on the road against a very good Vancouver team.”
Despite Saturday’s result, the Dynamo still sit bottom of the Western Conference with only 12 points, have recorded three wins, three draws and suffered seven defeats in their first 13 games of the 2016 season.
They now face a trip to arch rivals FC Dallas next Thursday before taking a two-week break from MLS play during the group stage of the Copa America Centenario.