Sir Humphrey is now called Rajnath Mathur and is played by Jayant Kripalani.
Ji Mantriji stars Indian actors and has been produced in Delhi in collaboration with BBC Worldwide.
It faithfully follows the original - which starred Sir Nigel Hawthorne and the late Paul Eddington - merely adapting the circumstances to Indian surroundings.
"The things that the British serial looks at - why the system never changes and nothing ever happens - all that is the same here, in fact it's exaggerated," said producer Smeeta Chakrabarti.
References to Russia in the UK version have been changed to Pakistan, cricket becomes football and the references to the EEC have been exchanged for the Commonwealth.
Flappable
New acronyms replace the dismissive TPLC (tin pot little countries) of the calculating advisor Sir Humphrey - played by Hawthorne in the original.
The minister character in the original - Jim Hacker, has been renamed Suryaprakash Singh and is played by the well-known Indian actor Farooque Sheikh.
Sir Humphrey is now called Rajnath Mathur and is played by Jayant Kripalani.
The idea behind Yes Minister was to expose the struggle between senior politicians and the bureaucracy - mostly it was the flappable minister who was at the receiving end of attempts to outwit him.
Mrs Thatcher was a fan. She said of the show: "Its closely-observed portrayal of what goes on in the corridors of power has given me hours of pure joy."
All the political spin and bureaucratic bungling of the original is imbued with Indian colour in Ji Mantriji.
Satire
Sir Anthony Jay, one of the creators of the original, said there was a universal appeal to the antics of bureaucracy.
"Indians do have a love of bureaucracy which isn't necessarily shared by the citizens they have to deal with," he said.
"And so do an awful lot of other countries."
Indian commentators have already queried the authenticity of keeping the Indian version so close to the original.
One journalist wrote: "Given the Indian political parameters, the bureaucrat is always at the mercy of the politician who keeps transferring him at will."
One of the stars of the show, Farookh Sheikh, said he wished it had gone further.
"I do feel the Indian political arena lends itself to satire more than most other countries and particularly countries in the west," he said.
Yes Minister was very popular in India when it was shown in English.
This is a unique project for the BBC as it does not usually adapt its series for regional markets.
"This is the first time BBC Worldwide has completely recreated a series for the Indian market or anywhere else in the world," said Mark Young, managing director of the India region for BBC Worldwide.
Rupert Murdoch's Star Plus channel licensed the series for pan-Asia transmission.