The Blue Jackets also kept goalie Sergei Bobrovsky even though he said before the season he will become an unrestricted free agent July 1. Bobrovsky made 106 saves and won all four games.
A month ago, the France-born Texier, 19, was playing in Finland. He got his first taste of North American-style hockey with seven games for the Blue Jackets' American Hockey League affiliate in Cleveland.
He played the final two regular-season games, scoring one goal, and Tuesday scored his first two playoff goals.
"For a young guy to come into a pressure situation and play like that and play like he did is pretty remarkable," forward Nick Foligno said.
Like Texier, there was a time Duchene couldn't have imagined being with Columbus.
"If you had told me last year the day I got traded to Ottawa (from the Colorado Avalanche) that I'd be here right now, I guess anything can happen," Duchene said. "You never know what to expect in this league. I'm very pleased to be here and excited for this group."