Messi finished with seven goals and would have won the tiebreaker on assists had he and Mbappé tied. Argentina won the epic final on penalty kicks (4-2) after the game finished tied at 3-3 in regulation.
Mbappe and Messi shared the lead on five goals going into the tournament finale between Argentina and France.
The race for the Golden Boot award, presented to the top goalscorer of the tournament, has gone into extra time at the FIFA World Cup 2022 in Qatar.
Mbappé, 23, is the first Frenchman to win the Golden Boot, though Just Fontaine scored a record 13 goals at the 1958 World Cup before the award was introduced.
Antoine Griezmann finished second in goals scored at the 2018 World Cup.
Mbappe is also just the fifth man to score in multiple World Cup finals and the second to score in back-to-back finals.
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