Kevin Mirallas surged forward and beat two Leicester players in the opening moments at Goodison Park and, after the Belgian was brought down, Davies pounced on the ball and poked it past Kasper Schmeichel and into the net.
The 18-year-old became the third player in the division to score after 30 seconds this season, after Chelsea's Pedro achieved the feat against Manchester United in October and Everton teammate Romelu Lukaku did the same against Bournemouth in February.
00:30 - Tom Davies' goal was the third scored after 30 secs in the Premier League this season after Lukaku's (Feb) & Pedro's (Oct). Rapid.— OptaJoe (@OptaJoe) April 9, 2017
The goal was Davies' second for the club after he opened his Toffees account with the third in a 4-0 win over Manchester City in January but Leicester hit back quickly through Islam Slimani and Marc Albrighton as three goals were scored inside 10 minutes of a Premier League game for the first time since Newcastle United and Arsenal drew 4-4 in February 2011.