The new supercomputer, which can perform more than four times the number of calculations per second than its predecessor, contains 1.8m core processors.
While the exact location of the infrastructure is a secret, the Azure supercomputing service is physically located in the south of England and is split across two data centres.
After recent hacks of British institutions such as M&S and Co-op, cyber security is on "everyone's thoughts" according to the Met Office's chief information officer Charles Ewen.
But he said he is happy with the security of the new system and is "very confident this is a big enhancement" on its predecessor.
"There's never been a more critical time for weather and climate intelligence," said Prof Simon Vosper, the Met Office's science director.
"We know the climate's changing, as we know the risks of hazardous weather.
"The supercomputer will unlock our ability to deliver those improved services where they're needed."
In February 2020, the UK government pledged £1.2bn for the project and said the old supercomputers would reach the end of their lives in 2022.
Mr Ewen said the three-year delay was predominantly caused by the Covid pandemic and its impact on the availability of silicon, while the Met Office highlighted the need to keep delivering existing services during the changeover.
"I'm very confident the Met Office is best placed in the world to be able to accommodate at scale emerging methods and techniques such as artificial intelligence," he said.
Microsoft's Azure cloud computing service is one of the largest providers of data centres in the world, and powers a large part of artificial intelligence infrastructure.
