Eakin took a pass from Jamie Benn on a 2-on-1 break and beat goalie Jaroslav Halak. The goal gave the Stars their first overtime victory in eight games this season.
Anders Lee tied it for New York with 2 seconds remaining in regulation when his shot from the left-wing boards hit defender Jordie Benn's skate and eluded goalie Kari Lehtonen.
Josh Bailey scored for the Islanders in the first period.
Brett Ritchie and Vernon Fiddler scored on third-period rebounds as Dallas rallied for a 2-1 lead. Lehtonen made 35 saves for the Stars, who were 0-4-2 in their previous six games.
Halak stopped 32 shots but the Islanders lost their second consecutive game.