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After a scoreless first period, James Desmarais fired a shot across Mike Clemente into the net for the Giants.
Taylor Dickin levelled and Cody Cartier scored the decisive goal just 43 seconds into the third period.
The Giants remain six points behind leaders Cardiff Devils in the standings, with one game in hand.
Storm kept Jackson Whistle busy in the first period and the Giants netminder again excelled in the second period.
Belfast player-coach Derrick Walser rattled the outside of the post in the second period and the visitors soaked up Giants pressure in the final minutes of the contest.
The Giants' next match is at home to Braehead Clan on Friday (19:00 GMT), the team they defeated 5-1 on Boxing Day.
After that, Coventry Blaze visit the SSE Arena on 2 January (16:00).
The Belfast Giants' unbeaten run of 10 games came to an end on Wednesday night with a 2-1 Elite League defeat by Manchester Storm at the SSE Arena.