Trump's overture to Putin was criticized by top senators in his own Republican party, who called Sunday's election a sham. Senate Majority Leader Mitch McConnell said there was a "lack of credibility in tallying the result."
When asked if Russia's election was free and fair, White House spokeswoman Sarah Sanders responded: "We don't get to dictate how other countries operate."
Putin won a landslide victory to extend his rule over the world's largest country for six more years at a time when his ties with the West are on a hostile trajectory.
"I congratulated him on the victory, the electoral victory," Trump told reporters at the White House while meeting Saudi Crown Prince Mohammed bin Salman.
"The call had to do also with the fact that we will probably get together in the not too distant future so we can discuss arms, we can discuss the arms race."
The White House later said there were no specific plans for a meeting.
Speaking of the arms race, Trump said: "We will never allow anybody to have anything even close to what we have."
Last week, the Trump administration hardened its stance on Russia by accusing Moscow of hacking into America's energy grid and approving the first sanctions on Russian entities and citizens for meddling in the 2016 US election. Russia denies interference in the election.
The United States also joined Britain, France and Germany in demanding that Russia explain a military-grade nerve toxin attack in England on a former Russian double agent.
Trump, who vowed to improve relations with Russia on taking office 14 months ago, is under pressure from Congress to take a tougher stance on Putin, his inner circle and scores of Russian oligarchs. Trump has not been highly critical of Russia in public, while maintaining there was no collusion between his presidential campaign and Russians.
"An American president does not lead the Free World by congratulating dictators on winning sham elections," McCain said in a statement.
The Organization for Security and Co-operation in Europe (OSCE) said there had been no real choice in Russia's presidential election and complained it had been marked by unfair pressure on critical voices.
The US State Department endorsed the OSCE's preliminary findings, said spokeswoman Heather Nauert, and called Trump's call to Putin "protocol."
The Kremlin said Tuesday's conversation had been broadly constructive and focused on overcoming problems in relations.
"The leaders spoke in favour of developing practical cooperation in different spheres, including on questions of how to ensure strategic stability and fight international terrorism," the Kremlin said in a statement.
Moscow and Washington are also at odds over crises in Ukraine and Syria, which Trump said would be discussed in a meeting along with the denuclearization of North Korea.
Putin and Trump agreed on the need to avoid an arms race and discussed "a possible high-level meeting," the Kremlin said.
Putin has struck a softer tone towards the West since Sunday, saying he has no desire for an arms race and would do everything he could to resolve differences with other countries.
Kremlin spokesman Dmitry Peskov told Russian news agencies separately that Putin and Trump had not discussed the poisoning of former Russian double agent Sergei Skripal and his daughter. London has blamed Moscow for the attack, a charge Russia denies.