The pair had already picked up Super Combined silver and Super-G bronze and then picked up their third medal with a time of 2mins 28.34secs over the two runs.
However, even then British duo were outdone by Slovakia's Henrieta Farkasova and Natalia Subrtova, who picked up their fourth gold by 5.34secs.
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Compatriots Kelly Gallagher and Gary Smith ended up fifth, with other GB representatives Millie Knight and Brett Wild seventh.
"I feel immensely proud of what we did out there," Fitzpatrick told BBC Sport.
"We definitely went out there and tried our very best. I was shattered by the end. We gave it our all.
We were completely relaxed and just went for it.
"After the fall we just had to be strong and overcome it and show our best skiing to the world and what we are made of."
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