Mbappé, who became France’s all-time leading goalscorer with two goals in a 3-1 win over Senegal in their opening World Cup fixture, scored twice against Sweden on Tuesday as they booked a showdown against Paraguay for a place in the quarter-finals.
The France captain ran Sweden's struggling defence ragged in New Jersey and lashed home his sixth goal of the tournament to draw level with Argentina superstar Messi at the top of the charts.
Mbappe extended his national record to 62 goals, having overtaken Olivier Giroud's previous French record of 57 with his historic brace against the Lions.
Meanwhile Messi has been in imperious form in the USA and has scored in all three World Cup games Argentina has played so far, including a hat-trick against Algeria and a brace against Austria.
Man City star Erling Haaland, playing at his first World Cup, tapped home in the 86th minute to score his fifth goal of the tournament to send Norway past Ivory Coast and setup a last 16 blockbuster against Brazil.
The prolific striker has scored in each of his last 13 competitive internationals, scoring 25 goals in that time, and overtook France's Ousmane Dembélé and Brazil's Vinicius Junior, who have each scored four goals in North America so far.
Elsewhere England captain Kane is also getting among the goals and overtook Gary Lineker's national record that stood for 36 years with his 11th World Cup goal against Panama, before overtaking Pele's record against DR Congo in the round of 32 with his 13th World Cup goal and his fifth strike of the tournament.
The Bayern stiker still has some way to go to match Just Fontaine's extraordinary record of 13 goals at a single World Cup, set in Sweden in 1958, but given the scintillating form of Mbappé, Haaland and Messi, few would be willing to bet against one of them mounting a serious challenge.
Yet in the most thrilling race for the Golden Boot in years, Mbappé and Messi are also leading the race to become the outright all-time goalscorer in the World Cup.
Mbappé has scored 18 World Cup goals in 18 matches, leaving him just one goal behind Messi's all-time tournament record of 19.
The Argentinian eclipsed Germany legend Miroslav Klose's long-standing pre-tournament record by scoring both goals against Austria, before extending his record with a trademark free-kick after coming off the bench in a 3-1 win over Jordan in their final group match.
It was Messi's 123rd international goal, leaving him second only to Cristiano Ronaldo's all-time record of 145, in 202 appearances, and marked a seventh successive World Cup game in which the eight-time Ballon d'Or winner has found the net, beating Fontaine and Jairzinho to set a new record.
While the race for the Golden Boot and overall record continues to gather pace, Mbappé played down the individual battles, insisting his focus is firmly on taking France all the way to the final.
"I think the goal, as I said, is to go as far as possible - to make it to (the final on) July 19th and come back here," he told reporters after reaching the last 16.
"We’re trying to win. We’re taking it one step at a time. Of course, the more goals you score, the higher you climb in the rankings - I’m not telling anyone anything new there.
"But I’m also convinced that Leo is going to score more goals, so I don’t focus too much on that. I’m more focused on the opponents we might face and how close we’re getting to our goal: the final."