1550s, "affirm, acknowledge openly;" 1590s, "make good, answer for," from Middle French avochier "call upon as authority," in Old French "call (to court), advocate, plead (a case)," from Latin advocare "call to" as a witness (see advocate (n.)).
Avouch, which is no longer in common use, means guarantee, solemnly aver, prove by assertion, maintain the truth or existence of, vouch for .... Avow means own publicly to, make no secret of, not shrink from admitting, acknowledge one's responsibility for .... Vouch is now common only in the phrase vouch for, which has taken the place of avouch in ordinary use, & means pledge one's word for .... [Fowler]
Related: Avouched; avouching.