It completed a four-point weekend for Adam Keefe’s side, but they were behind after the first 20 minutes through a shorthanded goal from Alexander Islamov.
Belfast came back as Sebastien Sylvestre equalised, only for Dylan Anderson to restore the Caps lead shortly after.
The visitors surged in front as Spiro Goulakos, Cole Jarrett and one each from Shields and Darcy Murphy had them 5-2 ahead after two periods.
Taylor Stefishen and Mike Cazzola brought Dmitiri Khristich’s men to within one, but Sylvestre’s second nudged Belfast ahead by two again.
Although Mike D’Orazio netted on the powerplay to bring the score to 5-6, they couldn’t capitalise any more as the Giants hung on for the points.