Sustr agreed to a one-year, $1.95 million contract Monday. He appeared in 80 games last season, finishing with three goals and 14 points.
Lightning general manager Steve Yzerman says Gourde signed a two-year contract worth $1 million a season. The 25-year-old played in 20 games last season, scoring six goals.
Monday is the deadline for clubs to extend offers to ensure the right of first refusal or draft choice compensation should players sign an offer sheet with another team.
Johnson and Palat were among those receiving qualifying offers before the deadline. Others were goalie Kristers Gudlevskis, defensemen Jake Dotchin and Slater Koekkoek and forwards Michael Bournival, Tye McGinn, Matthew Peca and Tanner Richard.