The Giants trailed for most of the match, which was heading for sudden-death extra time when hooker Danny Brough picked up a stray Crusaders pass and sent centre Leroy Cudjoe on a 50-metre sprint for the winning try.
That came after the visitors had been reduced to 12 men with the sin-binning of forward Jordan James for a professional foul.
Huddersfield, who had lost all three of their previous play-off ties, deserved their victory, having scored three tries to one, but will need to improve their performance in next week's semi-finals if they are to keep alive their hopes of a maiden Grand Final appearance.
If anything, the Crusaders earned most of the plaudits for a gutsy show which typified their efforts in reaching the play-offs 12 months after collecting the wooden spoon.
Full report and reaction in Monday's Yorkshire Post newspaper.