Sociedad had perhaps come into this fixture feeling the effects of their Champions League play-off win at Lyon earlier in the week, falling behind to Ferran Corominas' second-minute strike and looking sluggish in the opening half.
Elche, who were roundly beaten at Rayo Vallecano last week, could not hold on for the three points, however, and Mexican striker Vela grabbed a share of the spoils for Sociedad.
The visitors made several changes to the team that conquered Lyon in midweek. Antoine Griezmann and Carlos Martinez were injured and left out of the squad, while Xabi Prieto and David Zurutuza were left on the bench, meaning Esteban Granero made his first start since joining on-loan from Queens Park Rangers.
Corominas tested Real Sociedad goalkeeper Claudio Bravo in the opening minute, forcing a corner, which lead to Elche taking the lead.
A low cross from Edu Albacar wreaked havoc on the Sociedad defence and trickled over towards the far post, where Corominas bundled the ball home.
It was not the prettiest of goals, but with it being Elche's first goal in the Primera Division for 24 years, few of their supporters would have cared.
Alberto Botia was inches away from doubling the hosts' lead on 25 minutes, heading a free-kick just wide of the near post after Bravo failed to collect the cross.
With the visitors showing little signs of getting back into the game a quarter of an hour into the second half, coach Jagoba Arrasate turned to Prieto and Zurutuza, bringing them on for Ruben Pardo and Gonzalo Castro respectively within the space of 10 minutes.
Their arrival kick-started a Sociedad revival. First Vela made Elche goalkeeper Manu Herrera stretch to stop his shot and then the equaliser arrived, in which both substitutes played a key part.
Zurutuza found a gap in the defence for Prieto, who crossed towards Vela and although the former Arsenal man did not strike the ball cleanly, it ended up in the far corner of the net.
Granero nearly marked his full debut with a goal, but pulled a shot from the edge of the area just wide of the left hand post.
Vela then came even closer to grabbing the winner, striking the bar from 25 yards out, but the hosts rode out the 90 minutes to claim their first point of the season.