The ball broke to Davies, who dragged it out of the traffic before shooting into the bottom corner.
Four minutes before the end of an intense first half, Everton grabbed the lead. The winger drew the defenders before setting up his teammate for an easy finish.
But the reigning champions stormed back to lead after just 10 minutes courtesy of goals from Islam Slimani and Marc Albrighton.
In what was an nearly wildly open start to the game, Leicester equalised minutes later through Slimani, picked out by Demarai Gray in yards of space before slotting his finish coolly home after a characteristically quick counter attack.
Romelu Lukaku (Everton): It could easily have gone to Kevin Mirallas or Ross Barkley for all-action attacking displays, but Lukaku's finishing is vital to Everton's hopes of qualifying for Europe and on this evidence it's no surprise he's leading the way at the top of the goal charts.
Slimani had the first effort in anger in the second half on 50 minutes but his shot from the left side of the area took a slight deflection on its way into the side-netting.
Being on the road, it's safe to assume Leicester will play this game extremely conservatively, conceding most of the possession to Atletico.
His counterpart, Ronald Koeman, responded by introducing Gareth Barry to shore up the home side's midfield. We had great movement between the lines.
NWS confirms 2 tornadoes in Virginia
That number puts the Peach State in the lead for tornado reports , surpassing the usual leaders in the Great Plains. The weather service said that further surveys of the damage caused by Thursday's storms will be conducted Friday.
Lukaku scored twice in the 4-2 win against Leicester, taking his tally to 23 for the season, one more than Middlesbrough.
"When I saw the games in front of me, starting against Liverpool, I couldn't have imagined this". I've asked him still to be captain of the team and I think he changed his mentality in that.
"No, I am still not going to say that because I think we are never quite sure if other teams are capable of going on good runs", he said. "But the whole team performance was outstanding as well".
But Koeman also had a warning for Barkley, who is stalling on signing the contract he's been offered.
"Of course he has two more years' contract but everybody in football has his own ambition in football".
The Belgium global, who has been linked with a move away from the club after reportedly turning down a new contract, now has 23 top-flight goals this season, putting him four ahead of closest rival Harry Kane in the race for the Golden Boot. "Simple. It's not so hard in my opinion".
"I expect if we get six points out of the next two home games then we will fight for the fifth and the sixth place", the Dutchman added.