Messi, 38, overtook Klose to become the greatest goal scorer in World Cup history with a lovely effort from just inside the box in Argentina’s 2-0 win over Austria.
The legendary forward then went one better, taking his tally to an incredible 18 with his and Argentina’s second late in added time.
Argentina have won their two games 3-0 and 2-0, with Messi scoring every single one of their goals so far as they hope to retain the World Cup.
Klose has now congratulated Messi, telling Süddeutsche Zeitung: "Lionel Messi is the best footballer of all time for me! Congratulations, champion!"
The 48-year-old, who is now coach for Bundesliga 2 side Nuremberg, admitted that he had expected the record to be broken ahead of the tournament.
"That is perfectly alright, the record would be broken at some stage anyway, and I’d be happy for Messi to be the one to do it. I’m a huge fan of Messi – I always have been,” he said.