The transaction includes production, development and exploration sites off Denmark, Norway and west of Shetland.
Ineos said the move would make it the biggest private group operating in the North Sea.
The company has been looking recently to expand its upstream arm.
Last month it struck a $250m (£192m) deal with BP to buy the Forties pipeline system, which transports nearly 40% of the UK North Sea’s oil and gas production.
In the latest move, Ineos – which is controlled by British billionaire Jim Ratcliffe – will pay Dong just over $1bn, plus up to $250m in contingent payments.
About 440 staff will transfer from Dong on completion of the deal, which is expected in the third quarter.
Dong’s oil and gas assets include stakes in Ormen Lange, the second largest gas field in Norwegian waters, and Laggan-Tormore, a new gas field west of Shetland which came on-stream early last year.
Dong produced 100,000 barrels of oil and gas per day in 2016, down from 115,000 in 2015.
It is also estimated to have as much as 570 million barrels of commercial and potential oil and gas reserves across the Danish, Norwegian and UK continental shelves.
Mr Ratcliffe said the business was a “natural fit” for the group, which owns a major refinery in Grangemouth.
He added: “This business is very important to us at this stage of our growth plans and we are delighted with the expertise that comes with it.
“We have been successful in our petrochemical businesses, focusing on operating our assets safely, efficiently and reliably and we intend to do the same with our oil and gas assets.
“We are keen on further growth and already see lots of opportunity within this impressive portfolio when it transfers to Ineos.”