The tournament, which ends on Sunday with the final between France and Croatia at Moscow's Luzhniki stadium, has transformed "the perception that the world has about Russia," Infantino said.
"We did a good job together," he added, nodding to Putin.
"Thank you Mr President for your kind words," Putin said in English, which he rarely uses in public.
Putin told an audience of former soccer stars including Brazil's Ronaldo and Marcel Desailly as well as Russian government officials he was sad that the tournament was almost over but he also noted that it had helped improve the country's image.
"We cannot but feel sad that this spectacular football festival, which had given us so many happy moments, vivid impressions and new friends, is almost over," Putin said.
"Myths and stereotypes have been dispelled."
The month-long tournament has been held without major security incidents, while concerns about possible racist incidents and hooligan violence that were raised in the run-up have not been borne out.
Many fans from around the world have also spoken of a positive experience at the World Cup in Russia.
Putin said that Russia was eager to see World Cup fans return to the country after the tournament. Fans entered Russia without a visa, using a fan ID instead - a document mandatory to enter stadiums that proves its holder has been vetted and approved by the authorities.
"In Russia, we are always glad to see our friends, both old and new, now that we have so many more of them," Putin said.
Russian Deputy Prime Minister Olga Golodets, who oversees sport and culture, told Reuters this week that the country expected the World Cup to lead to a 15 percent increase in foreign tourism next year.
(Reporting by Gabrielle Tétrault-Farber; writing by Vladimir Soldatkin; editing by Polina Ivanova and Hugh Lawson)
