The go-ahead for the plant, which is expected to generate about 7 per cent of the UK's electricity needs and operate for 60 years, was confirmed in July but negotiations were still ongoing with French energy giant EDF about the exact funding arrangements with the UK Government.
The new plant will be built next to the existing Sizewell B, which is still generating electricity, and Sizewell A, which has been decommissioned.
In one of his last policy speeches as Prime Minister, Johnson said: "We need to pull our national finger out and get on with Sizewell C.
"That's why we're putting £700m into the deal, just part of the £1.7bn of government funding available for developing a large-scale nuclear project to final investment stage in this Parliament.
"In the course of the next few weeks I am absolutely confident that it will get over the line."
The announcement comes in the wake of record high energy prices in the UK, which is expected to cause significant financial hardship for many, and concerns over domestic energy security.
Ofgem's price cap is currently forecast to hit £3,600 in October and top £4,500 in January, which could see thousands of people having their power cut off and health endangered as they are unable to afford their household bills.
While he admitted nuclear energy was a more expensive option, Johnson said it would be "madness" not to go ahead with the project which would "fix the energy needs, not just of this generation but of the next".
"I say to you, with the prophetic candour and clarity of one who is about to hand over the torch of office, I say go nuclear and go large and go with Sizewell C."
He said that if the under-construction Hinkley Point C in Somerset was operating now "it would be cutting our national fuel bills by £3bn".
