USA media said said the man was discovered around midnight on Friday by a Secret Service agent, near the building's south entrance.
Tran confessed to jumping the fence of the White House.
Construction of the new 11-foot-7-inch fence, compared with the current 7-foot (2.13 m) barrier, is due to begin by next year, the Secret Service said in January.
"This is really troubling", said Jonathan Wackrow, an ex-Secret Service agent.
"I am a friend of the President".
Tran's younger brother Brian, 19, called him "troubled" after he was let go by an electrical engineering company.
Joseph Clancy, who replaced Pierson during Obama's administration, said last month he planned to step down in March, allowing Trump to name his own security chief.
He was living in his auto, the sibling added, before calling Tran a "very good brother".
Trump, who is spending the weekend in the nation's capital, was in the White House at the time of the intrusion.
Department of Homeland Security Secretary John Kelly has been briefed on the security breach.
Tran hid "behind a White House pillar before proceeding to the South Portico Entrance" at one point, the document states. Uniformed officers with the Secret Service arrested the person without incident, according to a statement.
When asked how he managed to get here, Tran replied: 'I jumped the fence, ' authorities said.
The backpack carried by the intruder was screened and searched as a precaution, and no hazardous material was found, according the statement. He was apprised of the situation, White House spokesman Sean Spicer told journalists Saturday.
"So the Secret Service has to really take a very hard look very quickly as to why weren't other defense measures alerted", Wackrow said.