The Faro side looked on course for victory when Livramento headed home a cross from former Portugal international, Jorge Ribeiro, early in the second-half.
Caldas levelled through Januário before Livramento restored Farense’s lead only for the home side to make it 2-2 thanks to a Pedro Emanuel goal.
Farense were reduced to ten men in extra-time as Neca picked up a second yellow card.
Five minutes away from penalties and local hero Pedro Emanuel got the winning goal for Caldas to send his team through to their first ever Portugal Cup semi-final where they will meet CD Aves.