Salas gave the visitors the lead in the 19th minute, but Enes Ünal equalised in the 68th before Martín Satriano put Getafe ahead five minutes later. Emmanuel Mangala added a third in stoppage time to complete a result that lifted the mood around the home side.
The decisive spell included a lengthy VAR review after the assistant referee had raised his flag for a possible infringement before Satriano's goal. José Bordalás was convinced the goal should stand.
"We had it very clear that it wasn't a foul. Justice was done. We were very clear on the bench, we had seen it, but on the monitor we reviewed it and it was clear. The referee got it right, his judgement prevailed. There shouldn't be any controversy," he said.
Bordalás was delighted with the way his players responded after falling behind against a Sevilla side he described as physically strong and well organised.
"We are very happy. We were coming to a stadium in spectacular form, with an incredible squad. We have to improve some aspects of our game, especially set-pieces. The team was playing well and then that goal arrived. It was like a weight on us.
"But the team knew how to react and turn around a match that wasn't easy. I'm pleased for the players. This has to be a first step towards improving and growing in LaLiga."
Sevilla head coach Luis García Plaza felt his side had controlled much of the contest but failed to take advantage of their opportunities before the momentum shifted.
"The difference in the match was finishing. When you are much better than the opposition for 60 minutes, you have to be 2-0 or 3-0 ahead. We had three or four very clear chances and, if you don't score, you give them life," he said.
"They pushed us, put their heart into it and have outstanding penalty-area forwards. I cannot say they didn't deserve the victory."
García Plaza also criticised the decision surrounding Getafe's second goal, arguing that his players had been conditioned to expect the referee to stop play when an assistant raises the flag.
"They raised the flag and the referee didn't blow. When they raise the flag, they always whistle," he said, while acknowledging that Sevilla had not been fortunate with recent decisions.
Bordalás, however, insisted the referee had been right to allow play to continue.
"All coaches tell their players not to stop because the person who makes the decision is the referee. If the referee doesn't blow, you don't have to stop. The final decision is always his."