The clan, under new coach Paul Gardner, got off to a great start and opened the scoring through Robert Farmer, before Ash Goldie doubled their lead before the intermission.
Two more goals in quick succession from Toms Hartmanis and Jade Galbraith gave the visitors a comfortable lead in the second period.
The Giants came back into the game when Will Colbert and Colin Shields hit the net, and the momentum seemed to be on their side after Matt Hanson caught Davey Phillips with a high stick, resulting in a four-minute powerplay.
But Braehead struck again with a shorthanded goal to effectively kill off the game, with Goldie rounding off the scoring at 6-2.
The Giants are back in action against Edinburgh Capitals at the Odyssey on Sunday.
Meanwhile, netminder Stephen Murphy has been named as part of Great Britain's squad for the Olympic final qualifiers in Latvia next month.
The 31-year-old has been absent for the past four last four Elite League games due to a groin injury but coach Tony Hand is hopeful his number one will be fit.
"Stephen Murphy picked up an injury playing for Belfast but he is on the mend and we are very hopeful he will be fit," said Hand.
"If he does not prove his fitness, there are two capable netminders on the reserve list to choose from."