Russia hosted the World Cup in 11 cities in June and July without any major security or discriminatory incidents.
The Republican president accused the media of distorting comments in which he gave credence to Putin's denials of election interference.
Trump's news conference alongside Putin drew fire from members of both political parties.
Russia was the target of almost 25 million cyber-attacks during the World Cup, President Vladimir Putin said, though he did not indicate who may have been behind the attacks.
Vladimir Putin welcomed His Highness the Amir and congratulated him for officially receiving the 2022 FIFA World Cup host's mantle.
His Highness the Amir extended congratulations to the Russian president and the Russian people for the success of organising the 2018 FIFA World Cup.
Macron and his wife Brigitte met the Russian leader at the Kremlin on Sunday afternoon before the match.
Gianni Infantino said preconceived notions of the country had been changed by the positive experiences of more than one million fans who had visited.
The Tass news agency reported that Bolton had discussed potential cooperation between the two countries' security councils with Yuri Averyanov, the first deputy secretary of Russia's Security Council.
Russia is spending more than $13bn on hosting football's showpiece, the most important event in the country since the 1980 Moscow Summer Olympics.