Zaragoza left-back Hector endured a nightmare league debut, sent off for two bookable offences after only 22 minutes to deal his team, already a goal down, a blow from which they would not recover.
Griezmann had opened the scoring after 11 minutes and Agirretxe doubled it shortly after the hour mark, with Zaragoza made to wait until deep in injury time for Apono to get their goal from the penalty spot.
The defeat means Manolo Jimenez's side retain their status as the La Liga's worst team at home this season.
Zaragoza perhaps enjoyed the best of the opening 10 minutes, but it was their only decent stretch in the game, and it would take until the 75th minute for them to attempt a shot on target.
Sociedad cashed in, opening the scoring when Carlos Vela broke through the middle and found Griezmann in plenty of space on the edge of the area.
The Frenchman looked offside, but the flag stayed down and slotted home with his second touch.
From then on Philippe Montanier's side took command of the game and their task was made considerably easier when 21-year-old Hector, promoted from the B team to make his league bow, was given his second booking inside nine minutes, both for fouls on Vela, and was sent off with 22 minutes played.
Things went from bad to worse for the home side 10 minutes later when Agirretxe doubled Real Sociedad's lead. The striker used his strength to fight off a challenge from Jose Fernandez and shot into the top corner of Roberto's net from just inside the area.
There was no way back for Zaragoza after the second goal, and the visitors began to dominate the game in every sense, maintaining 68 per cent ball possession and creating far more chances, with Roberto keeping the score dignified with saves from Alberto De la Bella and Griezmann in the final stages.
The only moment for Zaragoza's fans to cheer came in injury time when Helder Postiga won a spot-kick after being fouled by Markel Bergara. Apono converted a chipped penalty, but it was scant consolation for Jimenez's side, who are now looking over their shoulders at the relegation zone.
Real Sociedad, by contrast, are dreaming of European football.