Imanol Agirretxe nodded Sociedad ahead in the ninth minute at their Anoeta stadium in San Sebastian but Granada levelled six minutes later when Youssef El Arabi crashed a superb volley past home goalkeeper Claudio Bravo.
Carlos Vela raced down the left wing and crossed for Agirretxe to chest the ball cleverly into the net to restore Sociedad's lead moments before half-time.
However, after Agirretxe and Vela both struck shots against the frame of the goal relegation-threatened Granada mounted a late fightback and Recio threaded a 93rd-minute shot through a crowded penalty area to snatch a point for the visitors.
The result left Sociedad and Valencia, who won 4-0 at Rayo Vallecano on Sunday, level on 59 points in fourth and fifth place respectively with three games left.
Sociedad are above Valencia in the standings as they have a better head-to-head record but they have a tricky run-in to the end of the season and host Real Madrid at the end of the month.
Sociedad and Valencia are five points ahead of sixth-placed Malaga after their 0-0 draw at home to Sevilla on Sunday, while Granada are 16th on 36 points, four above the relegation places.
"It's a shame," Agirretxe said in an interview with Spanish television broadcaster Cuatro.
"We had those two shots against the post so we didn't have any luck in that respect," he added.
"Then we tried to keep the ball but we were the ones suffering towards the end."
Under French coach Philippe Montanier, Sociedad have played some of the most entertaining football of the season and are the only team apart from Real Madrid to have beaten leaders Barcelona.
Their golden period came in the 1980s when they twice won La Liga and reached the European Cup semi-finals in 1982/83, where they fell to eventual winners Hamburg SV.
More recently, they spent three successive years in the second division before winning promotion in the 2009/10 season. Fourth place would be their best result since they were second and qualified for the Champions League in 2002/03.
Granada were in the fourth tier of Spanish football as recently as 2006 and after winning promotion to the top flight in the 2010/11 season narrowly avoided relegation last term when they finished one place above the drop zone.
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