This famous piece first falls into the genre of the short story.  Because of its structure, the short story exists in its own fictional genre.  It is designed to be read in one sitting.  The exposition, rising action and climax receive the majority of the attention of the author - the falling action and resolution, and are often wrapped up within the climax.  O. Henry likes to make the climax/falling action/resolution present in the last line of the story - here, it is just short of the last line:

"Dell," said he, "let's put our Christmas presents away and keep 'em a while. They're too nice to use just at present. I sold the watch to get the money to buy your combs. And now suppose you put the chops on."

After this line, O.Henry adds a brief paragraph on the history of the magi, to put the actions of the characters in perspective.  This makes this particular short story fall into the subcategory of "Seasonal."  Clearly, O. Henry has set out to tell a Christmas story, not just a story of love and sacrifice.  If not, there would be no allusion to the Biblical magi.

The magi, as you know, were wise men--wonderfully wise men--who brought gifts to the Babe in the manger. They invented the art of giving Christmas presents. ...And here I have lamely related to you the uneventful chronicle of two foolish children in a flat who most unwisely sacrificed for each other the greatest treasures of their house.