The Giants had trailed 2-1 since the third inning before producing a run in the eighth when Belt sent home Angel Pagan with a two-out double into the right field corner.
In the 10th, Pagan (3-for-5) collected a one-out single against Adam Ottavino (1-3) and went to third when Marco Scutaro followed with another hit.
Then, with the outfield drawn in, Belt sent a fly ball to left field that landed behind Carlos Gonzalez, sending home Pagan with the game-winner.
Yorvit Torrealba knocked in a run for the Rockies.
Swim Week Miami Beach 2014 [PHOTOS]