The visitors struggled to get going as Southampton were clinical, taking a two-goal lead inside the first half through James Ward-Prowse and Che Adams.
With 15 minutes left on the clock, Lucas Digne was dismissed – the second red card shown to an Everton player in successive weeks – for an awkward challenge on the Achilles of Kyle Walker-Peters.
The Everton manager said: “We are going to appeal because it wasn’t intentional or violent conduct and I think it could be a yellow.
“It was an unfortunate contact and the red card was unfair.”
He added: “It was not a bad challenge. He was running behind and had the contact with the ankle of the opponent but no intention to hurt the player or use violent conduct.
“It was not the right decision.”
Ancelotti continued: “It was not a good day and not a good performance.
“We are here to talk about the first defeat and I think we have to move forward because of course we don’t like to lose games but in football it happens.
“It arrived after a fantastic start so we have to move forward with the same belief we arrived with here.
“It can happen.”
Southampton manager Ralph Hasenhuttl was full of praise for his side’s performance after their dominant victory.
The Saints boss told Sky Sports: “”It was definitely one of our best performances so far against a strong side. There was good game management and a good balance to our team.
“It was important that we kept the ball and made our opponents run, and we did this in a fantastic way. We’ve shown good steps forward in the past couple of games, and it’s necessary if we are to become a better side.
“It’s not important for me who scores, but what everyone does for the team. It’s important to make assists too, and our game management, and playing through the lines was fantastic today.
“We showed with the two goals how easy it is to score if you do it well. It was a dominant win as we limited their chances, and it was a good performance.”
