Trump entered a small elevator taking him to the third floor of the Apostolic Palace and, after a long ceremonial walk past frescoed corridors, shook the pope's hand at the entrance to the private study, which the frugal pope uses only for official occasions.
Francis smiled faintly as he greeted Trump outside the study and was not as gregarious as he sometimes is with visiting heads of state. Trump, seeming subdued, said "it is a great honor."
The two talked privately for about 30 minutes with translators.
Both men looked far more relaxed at the end of the private meeting, with the pope smiling and joking with Trump and his wife Melania.
Francis gave the president a small sculptured olive tree and told him through the interpreter that it symbolised peace.
"It is my desire that you become an olive tree to construct peace," the Pope said, speaking in Spanish through an interpreter.
Trump responded: "We can use peace."
"Well, I'll be reading them," Trump said.
Trump's softer stance on environmental regulations is at odds with Francis' view that climate change is caused mostly by human activity.
Parting promise
Trump gave the pope a boxed set of five first edition books by slain US civil rights leader Martin Luther King.
As Trump and the pope said goodbye at the door of the study, Trump told the pope: "Thank you, thank you. I won't forget what you said."
While his talks in Saudi Arabia and Israel were mostly friendly, the meeting between the head of the Roman Catholic Church and the thrice-married, blunt-spoken Trump had the potential to be a little more confrontational.
The pope said last year a man who thinks about building walls and not bridges is "not Christian," a sharp reprimand for Trump's vow to build a wall along the US border with Mexico.
"If and when the Vatican is attacked by ISIS, which as everyone knows is ISIS' ultimate trophy, I can promise you that the pope would have only wished and prayed that Donald Trump would have been president," Trump said during the campaign.
The Vatican also took a dim view of Trump's anti-Muslim campaign rhetoric, although he softened his tone considerably in a major speech in Riyadh.
Part of Trump's motivation for meeting the pope was to dramatise how the three major religions should rally against the threat from Islamist militants.
After the meeting, Francis held his weekly audience with the general public in St Peter's Square.