The Basque side impressed in their Champions League playoff first leg win at Olympique Lyon in midweek, when they won 2-0, but they were second best for much of the game at Elche.
Playing their first home game back in the top-flight, the highly-motivated hosts took the lead after only two minutes when a corner flashed across the area and Ferran Corominas stabbed in at the back post.
Sociedad were lucky not to concede again and struggled to get into the game until Elche's earlier efforts took their toll in the second half.
A cross from the right found Vela unmarked in the area after 74 minutes and the Mexico forward scuffed a shot inside the far post.
Former Arsenal striker Vela then curled a long-range shot against the crossbar near the end.
Sociedad play their Champions League playoff return leg at home to Lyon next Wednesday.
(Reporting by Mark Elkington, editing by Rex Gowar)
