Truong Trung Hieu was transiting en route from Liberia to Bangkok when customs authorities spotted him acting suspiciously, The Star (Kenya) newspaper reported.
They searched his luggage and found eight lion canine teeth and three claws and two warthog teeth inside a thermos flask.
Sniffer dogs had detected them, police said.
He is being held at the airport's police station pending an investigation.
The arrest comes on the heels of the 70th meeting of the United Nations Convention on International Trade in Endangered Species (CITES) where Kenya was taken off a list of countries of global trafficking concern, referred to as the gang of eight, which now includes Tanzania, Uganda, Malaysia, Vietnam, the Philippines, China and Thailand, the newspaper said.