The Mexican forward scored Sociedad's first goal just before half-time with a delightful chip and then set up Haris Seferovic for the clincher with 20 minutes remaining.
The win continued Sociedad's good form from last season, when the Basque club finished fourth; their highest finish in La Liga in 10 years.
Sociedad had seven shots on target to Getafe's two on Saturday and deservedly took the lead in front of their home crowd in the 42nd minute after Seferovic drifted in from the right with the ball at his feet and passed to Gonzalo Castro, who slid a first-time pass towards Vela.
The 24-year-old had gotten ahead of his defender and, as Getafe goalkeeper Miguel Angel Moya came off his line, Vela expertly lifted the ball over the Spaniard's head and into the net.
Vela turned provider in the 70th minute, releasing Seferovic in behind Getafe's right-back Juan Valera and the Swiss striker matched his team-mate's effort, chipping Moya from a tight angle to clinch victory for the hosts.
Sociedad will hope their win over Getafe gives them confidence heading into the away leg of their UEFA Champions League play-off against France's Lyon on Tuesday.
It was a good day for the Basque clubs in Spain as Athletic Bilbao also started with victory, beating Real Valladolid 2-1.
Markel Susaeta gave Bilbao the lead away from home just before the half-hour mark with a powerful drive, following a fine pass from Benat Etxebarria, but the hosts hit back just three minutes later when Patrick Ebert scored from Oscar Gonzalez's cut-back.
Bilbao scored the decisive goal five minutes into the second half thanks to a half-volley from Iker Muniain after a sliced clearance from a Valladolid defender had come back off the post.
In the third match of the opening day of the La Liga season, Valencia defeated Malaga 1-0 thanks to a 64th-minute goal from captain Ricardo Costa.
