The trade was announced just before the Sharks faced the Lightning.
A solid two-way forward, Moore has four goals and 19 points in 56 games this season.
"Dominic is an extremely versatile player that has performed in a lot of important playoff games the last two years," San Jose general manager Doug Wilson said in a statement. "We feel that he is a great fit in several areas and his veteran presence will benefit our team greatly."
Moore wasn't in the Sharks' lineup Thursday night.
San Jose is first in the Pacific Division but has lost in both of the last two seasons in the Western Conference finals.
In 41 career NHL playoff games, including 37 over the last two seasons for Tampa Bay (2010-11) and Montreal (2009-10), Moore had seven goals and nine assists.
The trade could be the first of several for Tampa Bay, which entered Thursday 13th in the Eastern Conference and is a long shot to make the playoffs.
Kubina has 110 goals and 382 points in 953 NHL games. The 34-year-old Czech has 11 points and is plus-1 in 52 games this season.
The NHL trade deadline is Feb. 27.