I searched this site and also searched etymology online and could find nothing about this question.
In BrE, the phrase 'being sacked' (Ngram BrE) is more popular and is more understandable as one would carry a sack home with any personal possessions, much as today people are seen with the ubiquitous cardboard box.
Where does the expression come from and why did it suddenly appear in the 1920s in the USA ?
Note: The Ngrams probably do not mean a lot, as there must be overlap with other meanings of 'being sacked' and merely querying 'being fired' would be useless.
Edit After Comment : The Ngram for 'given the sack' in BrE. Some correlation with 'being fired' in AmE.
Further Edit : The OED does not (that I can find) refer to losing a job but there is a considerable entry for 'fire' with regard to the discharge of a weapon so I am beginning to see that 'fire' from a job means to 'discharge' someone and the analogy is to weaponry.