Scotland Yard said officers are treating the incidents as linked and are keeping an open mind regarding motives.
No one has been injured in the three incidents and no arrests have been made, police said.
All the packages were A4-sized white postal bags containing yellow Jiffy bags and appeared capable of igniting a small fire when opened, the Met said.
Advice has been issued to transport hubs across London to be vigilant and report suspicious devices to police.
Officers first received a report of a suspicious package at The Compass Centre, a building near Heathrow Airport boundary, shortly before 10am.
A statement from Scotland Yard said: "The package was opened by staff at the building, causing the device to initiate.
"This resulted in part of the package burning."
-> Three explosive packages found at major transport hubs in LondonThe building was evacuated as a precaution and specialist officers attended the scene to make the device safe.
Police said the building remains closed as enquiries continuehttps://www.yorkshireeveningpost.co.uk/news/crime/grieving-mother-must-wait-to-see-her-son-s-killer-sentenced-after-crash-in-leeds-1-9631650.
A Heathrow spokesman earlier said that flights and passengers were not affected by the incident.
British Transport Police were later called to reports of a suspicious package in the post room at Waterloo station at 11.40am.
The package was not opened and specialist officers again made the device safe.
The station was not evacuated and trains continued to run as normal but cordons were in place outside on Cab Road.
Its owner, Network Rail, would not comment on the incident.
-> Drug dealer claimed cocaine was ‘gout medication’ after offering to supply undercover police in Leeds city centreOfficers were also called to a report of a suspicious package at Aviation House at London City Airport shortly after midday.
The package was not opened, the building was evacuated and specialist officers made the device safe. The building has since opened.
Docklands Light Railway services to the airport are running after an earlier suspension.
A spokesman for London City Airport said Aviation House was a staff-only building about three minutes from the terminal and no flights or passengers were affected.
Mayor of London Sadiq Khan tweeted: "Our thanks go to police, security, transport staff and all involved for their swift actions to keep our city safe."
Irish police are now helping counter-terror officers investigating the three small explosive devices sent to major transport hubs in London.
Security sources emphasised investigators were keeping an open mind, and added that the devices did not appear to be very sophisticated.
Following reports the packages had been sent with Irish stamps, a statement from Irish police said: "An Garda Siochana are assisting the Metropolitan Police with their enquiries."
Sky News correspondent Alistair Bunkall tweeted a picture which he said showed the package sent to Heathrow airport.
Read more: Grieving mother must wait to see her son's killer sentenced after crash in Leeds
The stamps appeared similar to some issued by the Irish postal service for Valentine's Day 2018, featuring a heart motif and the words "Love Eire N".
One of the images showed a partially-burned package addressed to Heathrow which appeared to have Dublin written in black ink in the bottom right-hand corner of the envelope, which Mr Bunkall said had caught fire.
The other picture showed a jiffy bag addressed to Waterloo, with its sender's address appearing to be Bus Eireann, Dublin.
The operator said police had not been in touch, with a spokeswoman saying: "Bus Eireann are currently not aware of this and we have no further comment."
The source of the images was not clear and their veracity has not been verified by the Press Association.