Trump had said during the election campaign he would be willing to withdraw US military stationed in South Korea unless Seoul paid a greater share of the cost of the deployment. There are about 28,500 US troops based in South Korea in combined defence against North Korea.
Park said the alliance between the two countries had grown as they faced various challenges over the past six decades, adding she hoped the ties would develop further.
She asked Trump to join in the effort to help minimise the threat from the North, which has carried out a series of nuclear and missile tests in defiance of UN Security Council resolutions and sanctions.
Trump agreed with Park and said: "We will be steadfast and strong with respect to working with you to protect against the instability in North Korea," the presidential Blue House said.
The official newspaper of the North's ruling Workers' Party said on Thursday the US wish for North Korea to abandon its nuclear weapons programme "is only a fantasy of a bygone era" and the policy of pressure and sanctions had failed.
"The only accomplishment of the Obama administration is that it is leaving behind for the new administration coming next year the burden of having to deal with a strong nuclear power," Rodong Sinmun said in a commentary.
It did not mention Trump by name. But Choson Sinbo, a pro-North newspaper published in Japan and controlled by Pyongyang, said: "Trump is well advised to learn the lesson of history from Obama's failure.
"Otherwise, the new owner of the White House will be met with the ashes of the calamity started by the previous owner."
The call between Park and Trump lasted about 10 minutes and Park said she hoped Trump would be able to visit South Korea soon, according to the Blue House.
There has been concern in South Korea that a Trump presidency will demand that Seoul sharply raise its share of the cost of maintaining the US military presence in the country.
Trump said earlier this year in various media interviews that he would be willing to withdraw US forces from South Korea and Japan but "would not do so happily".
"We get paid nothing, we get paid peanuts" for deploying the troops to South Korea," he said in an interview with CNN.
Under a five-year cost-sharing accord reached two years ago, Seoul agreed to contribute $867 million toward US military costs in 2014, about 40 percent of the total. The deal called for the amount to rise annually at the rate of inflation.
South Korea believes its share of the cost is much higher when the vast amount of land occupied by the US forces including a large area in central Seoul are considered.
Some members of parliament have suggested that the country has little choice but to consider nuclear armament if US forces are withdrawn while North Korea continues to develop nuclear weapons and missiles that could carry them.
South Korea's Defence Ministry spokesman Moon Sang-gyun said on Thursday the country has paid for its share of the cost of maintaining the US military and the contribution has been recognised by the US government and Congress.
South Korea and the United States have also agreed to deploy a Terminal High Altitude Area Defence (THAAD) anti-missile system with the US military to counter missile threats from North Korea.
South Korea has consistently said it had no plan to buy the THAAD system, which is built by Lockheed Martin Corp and costs an estimated $800 million a piece, that will likely add to the cost of maintaining the US military presence.
The two Koreas remain in a technical state of war after their 1950-53 conflict ended in a truce, not a peace treaty.
A man takes a selfie with a cut-out of Donald Trump during a US Election Watch event hosted by the US Embassy at a hotel in Seoul, South Korea
US President-elect Donald Trump pledged his commitment to defending South Korea under an existing security alliance during a phone call with South Korean President Park Geun-hye on Thursday, her office said.
