He balked in particular at Mattis' record in Afghanistan, where Trump has reportedly pushed to significantly cut the troop presence.
"As you know, President Obama fired him, and essentially so did I", Mr. Trump told reporters of Mr. Mattis' departure.
In his Cabinet meeting remarks with reporters present, Trump said he was in no rush to get US troops out of Syria but did not clarify a timetable.
Shanahan stepped into the job after Mattis resigned following Trump's decision to pull U.S. troops from Syria, a move that went against the advice of his most senior military and diplomatic advisers.
In his stinging resignation letter, Mattis made little attempt to hide his disagreements with Trump.
Mattis resigned after Trump's surprise decisions to withdraw all USA troops from Syria and half of the 14,000-strong contingent in Afghanistan.
The Pentagon's top financial officer David Norquist will soon take over as the acting deputy defense secretary, in succession to Patrick Shanahan, who took over as the acting defense secretary.
Trump similarly has criticized China on an economic and geopolitical level, but maintains he shares a warm relationship with President Xi Jinping.
"I'm not happy with what (Mattis has) done in Afghanistan and I shouldn't be happy", Trump said, as Shanahan sat by his side. Although Trump has declared victory there, critics warn Islamic State still has a toehold in Syria and could stage a comeback if USA forces depart. Like the Obama administration, President Donald Trump's government is concerned about what it calls China's militarization of disputed areas in the South China Sea and by its advances in high-tech weaponry. Before that, he spent more than 30 years at Boeing.
Seth Jones at the Center for Strategic and International Studies think-tank in Washington said Shanahan started the job from a somewhat weaker position, since he did not have the mandate that comes with being a Senate-confirmed defense secretary. "I have the greatest confidence in his abilities to lead a phenomenally talented team while performing the duties as Deputy Secretary of Defense", Shanahan said in a statement issued later on Wednesday.
Shanahan, a native of Washington state, holds a bachelor of science degree in mechanical engineering from the University of Washington and two advanced degrees from the Massachusetts Institute of Technology.
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