Bosnia took the lead in the game thanks to forward Jovo Lukic's header, his first international goal.
But a spirited second-half performance by the World Cup co-hosts resulted in a late equaliser by striker Cyle Larin.
"We left the initiative to Canadians, and we were playing in low blocks, and we were defensive," Barbarez told reporters.
However, both teams deserved the point, and the result was satisfactory, he said.
Barbarez reserved special praise for his team in the face of a raucous home crowd, who cheered every time Canada pressed forward.
"You were playing an opening match in front of a full stadium, it was a full house, and there were 80% of Canadian supporters. This is huge pressure, and it's a huge compliment for my team to have not succumbed to that pressure," he said.
He was also hopeful that 40-year-old striker Edin Dzeko, who was on the bench, would play in the team's next game against Switzerland in Los Angeles. Bosnia will then finish the group stage against Qatar.
A sharp goal-line clearance by veteran Sead Kolasinac denied Canada an equaliser early in the second half. Barbarez highlighted Kolasinac's experience, saying he expected such things from the defender.
"I expect him to hold the whole team together, to lead the team, and this is what he has done today in a top-notch way," he said.
The coach also took heart from the fight shown by his team and believes they will gain valuable experience from the match.
"I have to compliment the Canadian team. They were incredibly physical, incredibly intense. They had their plan. They knew how to pull it through," he said.