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It was Getafe's first league win over Spanish giants Barca since November 2011, a run of 16 games.
Koeman, who replaced Quique Setien at the start of the season, was less than impressed with the behaviour of Getafe full-ball Nyom.
"I told him Getafe coach Jose Bordalas that his number 12 Nyom had shown me a lack of respect, he had said two or three very ugly things to me which I won't repeat. He insulted me," Koeman said post-game.
"I told Bordalas that he needs to speak to his player, because we cannot tolerate that type of behaviour in modern football."
On Koeman's comments, Bordalas added: "I'm here to talk about football, not about controversy. I haven't spoken to the boy, I will tomorrow, but I doubt that Nyom would do that, I know him well.
"He's very competitive, but he's a very respectful person. I don't allow bad behaviour among my players but I doubt that Nyom insulted him."
Lionel Messi struck thepost and Getafe captain Djene Dakonam was spared from scoring an own goal by the crossbar, however – after starting his reign with back-to-back wins – Koeman has now seen his side pick up a solitary point from their next two outings.
Barca – who have their lowest tally of goals after four La Liga fixtures since 2015-16 with eight – had 73 per cent possession but managed just one on-target attempt asGetafeconceded 20 fouls in a physical contest.
After Mata converted a spot-kick following a foul by Frenkie de Jong on Dakonam, Koeman said: "I don't know if it was a penalty, it looked like they were looking for it and we fell into the trap. I don't know if we had VAR or not today.
"I don't like to talk about referees, but if you watch the match and see how many fouls they made and how many cards were given out, then you can make your own mind up.