The 48-year-old, who was sacked by Sevilla in December after a run of poor results in La Liga, joins the Basque club on a one-year contract.
Ziganda's departure was announced earlier in May, the coach lasting a single season after succeeding Ernesto Valverde at San Mames.
"For me it's an honour to be here, to defend those common values that we have," Berizzo, who had an operation to treat prostate cancer last year, told a news conference on Thursday.
"I appreciate all the confidence you've given me here, I hope to make Athletic better and have a defined style.
"I want to belong and be part of the Athletic philosophy, there is also a technical part and a sentimental part for the amount of people who have advised me to come here."
Berizzo previously worked with former Athletic boss Marcelo Bielsa during his spell as Chile coach and has also managed Estudiantes de La Plata, O'Higgins FC and Celta Vigo.