The Group C encounter was among six on Friday's program to launch the penultimate round of qualifiers in the CONCACAF region. The two top finishers in each of three groups advance to the finals qualifying round.
The Canadians overcame the expulsion of goalkeeper Lars Hirschfeld in the 64th minute as backup Milan Borjan entered and helped keep the Cubans from scoring the rest of the way.
Honduras is set to host Panama later Friday in the other opening match from Group C. Canada will host Honduras on Tuesday, and Cuba will visit Panama also on Tuesday before a break in qualifying matches until September.
Later Friday matches find Antigua and Barbuda at the United States and Guatemala at Jamaica in Group A and Guyana at Mexico and El Salvador at Costa Rica in Group B.