Sarabia capitalised on a Swiss error at the back to give Luis Enrique's side the lead in Geneva and that was enough for the visitors in a game that lacked clear-cut chances.
Spain, though, will be relieved to be up and running, this win coming after they were held to a 1-1 draw at home to Portugal and then needed a 90th-minute goal to salvage a 2-2 draw away at the Czech Republic last weekend.
Switzerland, meanwhile, succumbed to a third consecutive loss in the competition after they were beaten 2-1 away by the Czech Republic before being hammered 4-0 by Portugal, as Cristiano Ronaldo scored twice.
This was their first competitive loss at home in eight years, a run that stretched across 23 matches.
Spain are second in Group A2, two points behind Portugal while Switzerland are last, four points adrift of the Czech Republic.
"I think the balance of the game was very positive and that allowed us to take charge," said Luis Enrique. "We deserved the victory."
Spain needed a penalty shoot-out to beat the Swiss in the quarter-finals at last year's Euro 2020 and there was very little in this contest again, even if Spain were the more dominant side in possession.
Sarabia's goal came after an error by Eray Comert's heavy touch was also forced by some aggressive Spanish pressing, the loose ball seized upon by Ferran Torres before Marcos Llorente crossed for Sarabia to poke home in the 13th minute.
Neither side really looked like adding another and Switzerland's best chance came towards the end, as Haris Seferovic beat the offside trap in the 87th minute and looked to be through, only for Unai Simon to rush out and close the space.
Seferovic still managed to steer the ball across for Breel Embolo but he was 30 yards out and the chipped finish drifted wide.