The 40-year-old dog handler was held after her vehicle was searched, but police said the incident was not thought to be terror-related.
Scotland Yard said police recovered a very small amount of a substance, which was being forensically examined.
A Scotland Yard spokesman said: "On March 29, acting on information received, police stopped and searched a woman in a car park off Pudding Mill Lane.
"Her vehicle was searched and a 40-year-old woman arrested on suspicion of possession of an explosive substance and Class A drugs.
"She is in custody at an east London police station.
"The arrest was made by officers from the Olympic site support unit (OSSU) following information received.
"Another car was stopped and searched by the same unit on the M11.
"Nothing was found in the car and the driver was not arrested.
"Searches were carried out of residential addresses in Kent and London and no other substances were found.
"The investigation is being carried out by Newham CID.
"The Counter Terrorism Command is aware but the incident is not being treated as terrorism related.
"This incident did not represent a threat to the safety and security of the site."
The arrest came on the same day that building work at the £486 million Olympic Stadium was completed.
London 2012 chairman Lord Coe watched as Frankie Fredericks, a four-time Olympic silver medallist and the 1993 200m world champion, laid the last piece of turf on the infield.
It marked the start of the handover of the venue to the London 2012 committee.
A spokesman for the Olympic Delivery Authority said: "There are robust measures to ensure the safety and security of the Olympic Park.
"We are working with the police in their investigation. At no point has the safety and security of the Olympic Park been put at risk."
A spokesman for security firm G4S, which earlier this month signed a deal to be the official security firm for the Games, said: "Our canine services team is licensed to hold small samples of explosives for training purposes and are required to undertake rigorous training and follow strict operational processes.
"Our ODA canine services teams are highly effective because the dogs are able to regularly train using 'live' training aids. All dog handlers are fully vetted and operate to the highest industry standards."
"G4S take breach of operational processes very seriously and are assisting the police with their inquiries in relation to this incident."
