The London went down off Southend-on-Sea in Essex, killing 300 crew.
It is the first complete carriage to be raised from the ship, which is lying about 65ft (20m) below the surface.
The wooden mount, weighing about one ton, will be taken to the quay at Leigh-on-Sea. Historic England hopes it may provide clues as to why the ship sank.
Previous dives made by Historic England since 2014 have recovered parts of shoes, cabins, a sun dial and hand-made glass windows.
Five cannon from the 76-gun vessel were recovered before then. Two are with the Royal Armouries in Portsmouth and three are with a private collector in the USA.
Alison James, maritime archaeologist with Historic England, said: "Each carriage was made specifically for each gun, so we may be able to fit one of these five guns to it.
"We don't know if the carriage will be able to shed light on the explosion, but it's part of the wider excavation which we hope will explain it.
"The excavation is all done by touch because divers can usually see only 20cm [8ins] in front of them, so it's hard to see what is down there."
The carriage was lifted by a crane boat in about 30 minutes, but months of preparation had gone into the operation.
The ship, which was on its way to fight the Dutch, is resting on the sea bed in two main parts mainly covered in silt, or "Thames mud" as Ms James called it.
The finds are expected to go on permanent display at Southend-on-Sea Borough Council's Museums Service.
A gun carriage has been recovered from the wreck of a warship which exploded and sank in the Thames estuary in 1665.