Ultimately the source is from the Talmud:

Rabbis Ami and Asi would make a meal of the bread that was used for an eiruv, stating since it was used for one mitzva, let us use it for another (Talmud Shabbos 117b).

Rema (664:9) we put away aravoth and use them to bake matzo, for the reason aforementioned.

Since matzo baking has been commercialized, the custom developed to use it instead for the burning of the chametz.

Similarly, Tur (297) and Sh"A (ibid:4) re: besamim of havdalah, we use the hadassim left over from the lulav, for the same reason. Other examples include using the wax drippings from shul candles for Hanukkah (Bach 673), and using leftover tzitzit as a sefer bookmark (Magen Avraham 21:1).