The Capital side went down in straight sets 3-0 in Ayrshire and Ragazzi will meet Bon Accord in the final next month.
After losing the first two sets, City took charge late in the third set and led 21-17, needing just four points to take it to a fourth set.
But Ragazzi rallied and eventually came through a tense final few points 28-26 to seal their final place.
Bon Accord beat Glasgow Mets in the other semi-final 3-1 to reach their first final.
St Andrews University reached the women’s final – the first student team to qualify for a major final – after beating NUVOC and will face hot favourites Su Ragazzi in the final, Ragazzi beating Troon in the other semi-final.
There was some consolation for City of Edinburgh in the Under-16 Junior finals yesterday as they took both the boys’ and girls’ titles.