With 118 minutes on the clock, both nations looked like they would be forced to endure the drama of a penalty shootout in order to book their quarter-final place, but Angel di Maria slotted home to seal Argentina's progression.
Sabella was in the dugout when La Albiceleste beat England on penalties in the 1998 edition of the tournament, and the coach is thankful to be spared the same anguish this time around.
"The first half was quite close and Switzerland got two good chances to open the scoring. But after the interval, we started to show our credentials and we deserved to win," he told reporters.
"It was in 1998 when I was an assistant coach that we had to go through penalty kicks against England, but fortunately we could avoid that this time by winning after 120 minutes.
"We have to rest now ahead of the quarter-finals. We put in a great effort and got a just reward for it."
Sabella's side will face either Belgium or USA in Saturday's quarter-final.