I think we may to know how many innings the game lasted.
Not sure if Fed rules are the same as OBR on this ... but OBR says that in a game of 6 innings or more, the starter must pitch at least 5 innings to get the win. So maybe the reliever here gets the win and not a save.
Midlo, again in looking at OBR (which may not apply here), the part you mention about "have to have pitched effectively" is not mandatory for qualify for a save. It's part of a group of 3 conditions, only 1 of which must be satisfied.
There is a provision that if a starter can't get the win due to not enough innings pitched and his team then remains in the lead throughout, the win goes to the relief pitcher whom the scorekeeper deems to have been the most effective pitcher.
So your 14-13 example would create an interesting dilemma for the scorekeeper if the starter just went an inning or two and there was only one relief pitcher.
I'm interested to hear from others who are more knowledgeable than I am of Fed rules on the issue of the save.