Davis struck first before Jordan Ayew found an injury-time equaliser for the Foxes.
The defender later said: "We came in frustrated. I think we could have had a penalty. Everyone makes mistakes, we're human but it should have been given. We have to watch our heads after that. We're just disappointed."
On Kalvin Phillips' red card, he continued: "If you're playing against 10 men then you get that momentum swing. I think he's disappointed it happened and in his head he probably thinks he needs to be cleverer but it happened.
"I'm buzzing that is my first goal but I would have been happier with the three points. We want to get points on the board. I was just in the right place at the right time. I've always wanted to score in the Prem.
"Everyone came in with clear heads and worked hard on the Monday morning after last week. We'll watch it back and improve."