In a statement, Hinrich Woebcken, CEO of Volkswagen America, said: "Final approval of the 2.0L TDI settlement is an important milestone in our journey to making things right in the United States, and we appreciate the efforts of all parties involved in this process.
"Volkswagen is committed to ensuring that the program is now carried out as seamlessly as possible for our affected customers and has devoted significant resources and personnel to making their experience a positive one."
The settlement allows owners of affected vehicles to sell their cars back to Volkswagen for a pre-scandal trade-in price, as well as penalty-free lease terminations.
There will also be compensation of up to US$10,000 ($13,100) for current owners and lessees, as well as people who sold their cars before news of the Dieselgate affair broke.
Depending on the number of owners who opt to keep their cars, the overall cost to Volkswagen could be significantly lower than the headline figure of almost $20 billion.
As part of the deal, Volkswagen will fund environmental programs to the tune of US$2.7 billion ($3.6 billion) to mitigate the effects of the extra NOx (oxides of nitrogen) that its cars have illegally put into the atmosphere.
It will also spend US$2 billion ($2.6 billion) to improve electric vehicle infrastructure and access.
The large settlement covers the roughly half-a-million vehicles sold in the USA, under the Volkswagen and Audi brands, powered by a 2.0-litre turbo-diesel that's equipped with a defeat device that allowed it to illegally detect and pass laboratory emissions testing.
While today's approval finally provides closure to some of the largest pieces of litigation facing the Volkswagen Group, there are still a number of other outstanding legal issues to be dealt with.