Darren Mattocks scored in only the fourth minute to give the Canadian side the advantage.
However, Mike Magee equalised in the 69th minute following a David Beckham corner - and four minutes later Landon Donovan converted a penalty he had won to settle the contest.
The result earns Los Angeles a Western Conference semi-final double header against San Jose.
Donovan later admitted a halt-time chat inspired their fightback.
He told www.mlssoccer.com: "We're passionate and we don't want to lose.
"We said at half-time, 'Let's stay positive,' because it is frustrating when you work that hard and the goal doesn't come.
"We've got a number of guys that have the experience and know how to play in games like this, even when we're down a goal.
"A lot of teams would probably get pretty nervous and out of sorts if they gave up that goal (to Mattocks), and I thought our willingness to keep doing what we were doing was really good.
"To come back against a team that was content to defend with 11 guys behind the ball, I thought it was really impressive that we were able to break them down."