Gordon McRorie added the rest of Canada's (1-0) points on two penalty goals and two conversions.
Matias Arocena hit on two penalties for Uruguay (0-1).
Uruguay opened the scoring, going up 3-0 on Arocena's first penalty goal, but McRorie responded with a penalty of his own to draw even at the 10-minute mark.
Barkwill's try and a conversion from McRorie put Canada ahead 10-3. McRorie's second penalty gave the Canadians a 13-3 advantage at the half.
Arocena's second-half penalty goal brought Uruguay to within seven points. Gilmour's try and a conversion from McRorie sealed the victory for Canada.
Earlier in the day, four-time defending ARC champion Argentina (1-0) opened the tournament with a 41-14 victory over the United States (0-1).
Dan Isaack had three tries to lead the Argentines, while Matias Diaz, Rodrigo Baez and Patricio Fernandez also scored.
The four teams will play again on Wednesday, with Canada taking on the U.S., and Uruguay challenging Argentina.