Olivier Occean gave the visitors the winner with a header after 54 minutes.
Canada were then reduced to 10 men when Hirschfeld was shown the red card after 63 minutes for handling the ball outside the penalty box.
"I came out pretty aggressively and basically looking to clear it with my head and play it out to the left back when it bounced it popped up really high over me," Hirschfeld explained.
"It was a desperation thing, either let it get up over me and into the box and let them have a go at it."
Cuba, who were unlucky not to score early in the second half when Alberto Gomez's shot just missed the target, pressed hard for an equaliser but were unable to find a way past replacement goalkeeper Milan Borjan, who came on for Tosaint Ricketts when Hirschfeld was sent off.
Canada, bidding to reach the World Cup for the first time since 1986, converted their best opportunity when Occean headed David Edgar's cross past Cuban goalkeeper Odisnel Cooper on a rough pitch that made it hard for the teams to control the ball.
"It was difficult for the players today, very, very difficult," Canada coach Stephen Hart said.
"Cuba are going to take points from anybody that they play here. A lot of teams that have a passing game and a passing pattern are going to struggle."
Canada and Cuba are among 12 teams through to the third round of CONCACAF qualifying. The top six will advance to the final round with the final three advancing automatically to Brazil in 2014.
Canada's next game is against Honduras in Toronto on Tuesday while Cuba are away to Panama.