Things went from bad to worse when David Luiz tripped Roberto Pereyra inside the box and the substitute converted the penalty to level the scores.
Head coach Emery refused to criticise his players, while captain Granit Xhaka claimed they had been “scared” in the second half.
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But Sokratis, signed from Borussia Dortmund last summer, said: “I think that I cost the game for the team. It was my mistake for something simple that I don’t have to do.
“I think this was the problem why we lose the three points. This is all about our responsibility and we have to do better. It was my mistake and today without my mistake, we would take the three points.
“I don’t want to speak about others. Today it was my mistake and this cost the game. Nothing else. The other guys tried their best. I don’t have to do this mistake.”
Watford had 31 shots on goal during the game as they dominated after the interval, with Sokratis admitting Arsenal were not up to scratch in the second 45 minutes.
“Of course. I apologised because today I cost the game for the team and I have to work harder,” he added.
“I know up to this moment I didn’t have a mistake, I didn’t do nothing. Everything was near to perfect. We didn’t have the same pace in the second half as in the first half.
“But if we didn’t do these mistakes, I don’t think we draw. Of course they had other chances to score more, but if we didn’t do these mistakes then nothing happens.”