Arrasate, a coach from the club's youth ranks, was part of Philippe Montanier's backroom staff last season.
Frenchman Montanier rejected the chance of a contract extension with the San Sebastian-based club last month despite leading them to fourth in La Liga and into the Champions League play-offs for next season. He has since joined Stade Rennes.
"Jagoba has a complete understanding of La Real from the inside and he meets the characteristics that we were looking for," Aperribay said.
"Jagoba has the ability to take the reins of the first team. He can continue what Philippe started."
Sociedad were one of the most exciting teams to watch in La Liga last season, winning plaudits for their dynamic playing style with a team made up largely of home-grown players.
They ended champions Barcelona's record-breaking unbeaten start to the season with a 3-2 home win in January and were the first team to beat third-placed Atletico Madrid at the Calderon in March.
They scored six times in a defeat and a draw against Real Madrid and qualified for Europe's elite club competition for the first time in a decade.
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