Porto midfielder Eustaquio swept in a low finish on 85 minutes to seal victory for Canada after Jamaica had fought back to level through Shamar Nicholson earlier in the second half.
Saturday's quarter-final was initially due to have been played on Friday but was postponed after torrential rain left the National Stadium pitch waterlogged.
Instead, the first leg game kicked off in bright sunshine at 10.30am local time on Saturday before a virtually empty stadium. Canada took the lead in first-half stoppage time after a sustained period of dominance.
Real Mallorca winger Cyle Larin lost his marker just inside the penalty area and fired a low cross across the six-yard box where an unmarked Jonathan David was waiting at the far post to tap in. But Jamaica hit back early in the second half after a defensive lapse by the Canadians.
Daniel Johnson threaded a low pass into Nicholson on the edge of the area, and the France-based striker punished sloppy marking to finish emphatically for his 11th international goal.
Jamaica's celebrations were briefly put on pause after the goal was initially ruled offside, but VAR overturned the decision after replays showed Nicholson was clearly onside.
The Reggae Boyz looked as if they would hold out for a draw, but five minutes from time, Richie Laryea made a surging run down the right and cut back a cross into the path of Eustaquio, who made no mistake with the finish. The second leg of the quarter-final takes place in Toronto on Tuesday.Read on besoccer.com