The trade took place just minutes before the two teams squared off Thursday night.
Moore had played in all 56 games for the Lightning this season, recording four goals and 15 assists.
In 507 career games with the New York Rangers, Pittsburgh, Minnesota, Toronto, Buffalo, Florida, Montreal and Tampa Bay, Moore has 67 goals and 112 assists.
"Dominic is an extremely versatile player that has performed in a lot of important playoff games the last two years," Sharks executive vice president and general manager Doug Wilson said. "We feel that he is a great fit in several areas and his veteran presence will benefit our team greatly."