The series came down to a decisive fifth rubber after Bouchard opened the day with a 6-4, 7-5 victory over Tsurenko followed by Fichman's 6-4, 7-6 (4) loss to Svitolina on a indoor clay court at the Sport Club Meridian.
"I wasn't worried about losing my singles match because I knew I could get another chance in doubles," said Fichman. "As a player, all you can ask for is an opportunity and we made the most of it today."
The 19-year-old Bouchard took early control in the first rubber, jumping out to a 5-2 lead in the opening set. After falling behind 5-3 in the second set, the Westmount, Que., native reeled off four straight games with some impressive winners to come away with a straight sets victory.
"It was important to stay in the points because she (Tsurenko) can hit some amazing winners, but she can be erratic sometimes so you have to be ready for that," said Bouchard. "I am happy that I was able to stay with her mentally and be aggressive when I had the chance."
It was Bouchard's fifth win out of the last six matches representing her country in Fed Cup play.
The roles were reversed on Saturday when Bouchard lost the opener to Svitolina before Toronto's Fichman levelled the series 1-1 with a win over Tsurenko.
Canada becomes one of eight teams to qualify for World Group II in 2014. Canada last competed in World Group II in 2011.
The Canadians advanced to the World Group II playoff in February after winning all four of their ties to finish first in the Americas Zone Group I event in Medellin, Colombia.