Arrasate was assistant to Philippe Montanier in the 2012/13 campaign as Sociedad finished fourth in the La Liga to secure a Champions League qualifying berth.
Montanier announced last month that he would be leaving the Basque club at the end of the campaign after turning down the offer of a new contract.
Speaking at a press conference this afternoon, club president Jokin Aperribay said: "Jagoba Arrasate is going to be the next coach. He is the one to continue the strategic plan we have worked to for years."
Former France boss Laurent Blanc was linked with the role, but Aperribay insisted the club had only two names on the shortlist to succeed Rennes-bound Montanier - Arrasate and Gerardo Martino.
With the latter having reportedly rejected an approach in favour of remaining in his native Argentina as Newell's Old Boys coach, Sociedad turned to Arrasate.
And Aperribay claimed he did not consider it a risk to hand the reins to the less experienced man.
"When the door closed on Montanier we thought of two names - Tata Martino and Jagoba Arrasate," he said.
"We think we will start next season without doubts, and with a perfect knowledge of what we have.
"It is not a gamble. We want to continue the sporting path, and this was an opportune moment to do that."