Moore scored a power-play goal with less than 2 minutes to go in the second period and another 6 minutes into the final frame, giving the Lightning a 5-2 advantage.
The Bolts, however, ended up needing both of those tallies. Brendan Morrow narrowed the advantage to 5-3 only 13 seconds after Moore's second goal and James Neal pulled the Stars to within one with a minute left in the contest.
But Tampa Bay goalie Mike Smith held firm in the final seconds and the Lightning emerged with the victory, improving to 4-1-0 to start the season.
Steve Downie and Steven Stamkos added first-period scores for the victors.
Toby Petersen and Adam Burish found the net for Dallas, which was handed its first loss of the season (4-1-0).