Sauce reaches Latin salsa, the feminine form of salsus, salted. That participle comes from sallere, to salt, built on sal, salt.
This sequence mixes word formation with borrowing. Anglo-French and Middle English lie between the Latin and modern English stages; they are compressed. Salsa and salsus are grammatical forms, not separate borrowing languages.
Sources
- Merriam-Webster: sauce
Noun Word History supplies sal, sallere, salsus/salsa and the route through Anglo-French and Middle English.
- Online Etymology Dictionary · sauce
Meanings of the older forms on the word path.