Cozens sustained the injury during a 3-on-3 scrimmage on the final day of Sabres development camp.
"The puck was going to the wall and I tried to chip it around the [defenseman] and dodge him to the inside and he kind of clipped me a little bit," he said. "It was awkward, and I fell right down on my thumb, perfectly straight on my thumb, all my weight went down on it, and it shot it back.
"The moment I landed on the ice ... I felt, 'Oh boy, let's see which way this thing is sticking out.' I knew it was dislocated, I could just feel it. And I took my glove off and saw it and said that was pretty gnarly looking."
With playing in the Prospects Challenge, which includes top prospects from the Pittsburgh Penguins, Boston Bruins and New Jersey Devils, now a possibility, Cozens said he hopes to have a full training camp to show the Sabres he can play in the NHL this season.
"I believe I do have a shot at making the team," he said. "Obviously not too many 18-year-olds make it, but I feel I have the size, I have the skating, I have the strength to be a guy who can step in at a young age. Obviously, I want to play when I'm ready and if they think I need another year to develop then I'll be happy with that, I'll be happy to go back to Lethbridge (Western Hockey League) and play there. But my goal now is to make the (NHL) roster."