The Scottish hosts were 3-0 ahead by end of the first period on Saturday night after goals from Stephen Gunn, Sebastien Thinel and Richard Jackman.
David Rutherford pulled one back for the Giants but Justin Fox restored Fife's three-goal advantage.
Rutherford completed his double before Chase Scaber ended hopes of a Belfast comeback in the third period.
The two teams will meet in the Challenge Cup return game at the SSE Arena next Saturday night.
Elite League action gets underway for the Giants 24 hours later with a trip to take on Coventry Blaze.
Belfast Giants made a losing start to the 2016/17 campaign with a 5-2 defeat by Fife Flyers in the Challenge Cup.