Despite the wild shooting gallery in front of Cuban goalie Odelin Molina, Canada managed only one goal in 72 minutes before scoring twice in five minutes to emerge a 3-0 winner before 17,712 on Friday.
Advancing to the final round of CONCACAF qualifying is still well within Canada's reach. A tie or win Tuesday in Honduras will be enough.
Panama tied visiting Honduras 0-0 in the other final Group C game.
That means Panama (3-1-1) and Canada (3-1-1) are tied with 10 points. Honduras (2-1-2) has eight while Cuba (0-5-0) has zero.
Panama hosts Cuba in the other game Tuesday.
Canada could even lose in Honduras and advance, in the unlikely event that Cuba beats Panama and the goal difference goes Canada's way.
The Canadians will be facing a Honduran side in need of a win to advance.
The top two in Canada's group advance to the final six-team round of qualifying in CONCACAF, which covers North and Central America and the Caribbean.
From there, three will advance to the 2014 World Cup in Brazil. A fourth CONCACAF team will take part in an intercontinental playoff with the Oceania winner to see who joins them.
Asked what a fair scoreline would have been Friday, Canadian midfielder Atiba Hutchinson paused before saying "I think we could have scored at least seven goals today, we had so many chances.
"Obviously we only had three but there could have been a lot more. And that's one of the reasons why players had their heads down a bit. Because we knew we could have had a lot more goals."
Captain Kevin McKenna was disconsolate as he left BMO Field.
"We're just not finishing our chances. It might cost us in Honduras," he said, knowing that goal difference is a tiebreaker.
"It's in our hands, obviously, if we draw or win the game (in San Pedro Sula)," he added. "But it would have been nice if we had taken a lot of pressure off ourselves by scoring more goals tonight and we didn't do that."
The Cubans dressed just 11 players for the game, meaning their bench was empty. Down the field, Canada had 10 substitutes dressed.
After the game, Cuban coach Alexander Gonzalez was reluctant to talk about the missing players other than to say through an interpreter that some had left the team.
Pressed on the issue, he said he came to Canada with 15 players. He arrived at BMO Field with 11.
FIFA rules says a team should consist "of not more than 11 players, one of whom is the goalkeeper. A match may not start if either team consists of fewer than seven players."
Hart knew the Cubans were down players but opted not to tell his team about it before kickoff.
For Hart and his players, playing against a team with no substitutes was a new experience.
"It was strange," said Hart.
And Cuba was reduced to 10 men in the 70th minute after striker Roberto Linares was set off on a challenge that crumpled captain Kevin McKenna.
Three minutes later, Canadian striker Olivier Occean was sent off for a melee that ensued after Will Johnson's headed goal made it 2-0. That means Occean will have to miss the Honduras game.
Whatever the numbers, it was one-way traffic on the field and Canada — ranked No. 61 in the world — should have been ahead by a touchdown after 45 minutes.
The chances were there, but the finishing wasn't for most of the game. It's a familiar story for Hart who saw his team put 14 shots on target to Cuba's one (Canada led goal attempts 35-7) while holding 9-0 edge in corners.
Hart said he thought his team began to doubt itself and lose discipline when the goals did not keep coming in the first half.
Asked what he said at halftime, Hart replied:"The one thing I didn't want to say is keep on doing what we've been doing."
Instead he preached patience and ball movement, to see if he could stretch a tiring Cuban team and find room behind the visitors' defence.
Tosaint Ricketts scored for Canada in the 14th minute. And until Johnson's goal padded the lead with just 20 minutes remaining, the Cubans were one kick away from evening the score and silencing the enthusiastic crowd at BMO Field.
Fullback David Edgar made it 3-0 in the 78th minute with a sweet volley coming in off the flank.
"I would have liked to score six," Hart said later. "But to be honest we scored some goals, we won the game.
"And I told the players our objective should be to win the game. And what happens happens. And we did that, we scored some goals. And I'll take it."
McKenna was far gloomier, especially about the missed chances in front of goal.
"What can you say? It's been like that in Canadian soccer for a long time. I don't know."
"For me it's not good enough," he said of Friday's performance.
The Cubans, ranked 146th in the world, are just playing out the string, glued to the bottom of their four-team group.
The Cuban coach said later he was working in new talent into his squad, players chosen for "a passion and a love of the game." He noted that two of his starting 11 played injured, included one with five stitches in his leg.
Once the game started, Canada's subs and coaches sat in a line of chairs in front of the dugout. The Cuban coaching staff opted to sit in their dugout with only the coach stalking the sideline.
Ricketts finally scored for Canada in the 14th minute, set up by a beautiful interplay between Ante Jazic and Simeon Jackson. Jazic's cross eluded Hutchinson but Ricketts was there to tap it in.
The first half ended with a flurry of Canadian attacks. Canada outshooting Cuba 21-2 (9-1 in shots on target) in the half. The home side had nine corners to Cuba's none.
Cuba regained some of its poise in the second half but Canada still held the upper hand, coming at the visitors in waves. At times, it was a shooting gallery in front of Molina and more than one Canadian player should have left the ground with a big stuffed animal.
Canada is bidding to reach the final round of qualifying in the region since 1997 when it finished last in the six-man group with a 1-6-3 record.
The Canadian men have only qualified for the World Cup final once — in 1986.
Canada opened this round of qualifying with a 1-0 win over Cuba in Havana in June. The Canadians tied Honduras 0-0 in Toronto next time out.