Nirra Fields added 12 for the Canadians, who went a perfect 3-0 through the preliminary round. Miranda Ayim had eight while Kia Nurse finished with seven.
Achonwa, a rookie with the WNBA's Indiana Fever, was making just her second appearance for Canada since tearing her anterior cruciate ligament last year in what would be her final game of a standout college career for Notre Dame.
The Canadians took a 12-point lead into the fourth quarter but Cuba used a 12-2 run to pull to within two points with just over four minutes to play. But when Miah-Marie Langlois found Achonwa alone under the basket for an easy two points, it put the Canada up by five with 2:53 to play.
Cuba's Leidys Oquendo was fouled on a three-pointer with 12 seconds left but missed one of her free throws, allowing Tamara Tatham to secure the victory with a basket with six seconds remaining.
A sold-out crowd of an estimated 3,300 fans — including men's team members Sim Bhullar and Jamal Murray and retired national team star Teresa Gabriele — took in the game at Ryerson's Mattamy Athletic Centre, making for one of the largest home crowds the Canadian women have ever played in front of.
The Canadians trailed for the first time this tournament early in the opening quarter, but took a 20-18 lead into the second, and a 40-35 advantage into the dressing room at halftime.
Canada shot just 2-for-10 to open the third as Cuba pulled even at 45-45 midway through the frame. But the Canadians dug in down the stretch and a rebound and bucket at the buzzer by Michelle Plouffe sent Canada into the fourth up 60-48.
The Canadians rolled over Venezuela 101-38 in their opener before holding off Argentina 73-58. The Cubans, meanwhile, beat Argentina 68-55, then Venezuela 73-62.
Canada has never won basketball gold at the Pan Ams, on either the men's or women's side. The women have won one silver, in 1999 in Winnipeg, and three bronze.
The Pan Ams are just of two huge tournaments on home turf for the Canadian women this summer. They'll play the FIBA Americas Olympic qualifying tournament next month in Edmonton.