Bobby's Sweet Centre on Roundhay Road first started selling hot food in 1974, but will now shut after its owners decided to retire.
The kebab house was set up by Indian immigrant Ram Murti Thukral, who moved to Leeds in 1969 and originally opened the shop to sell sweets and groceries from his home country.
The premises had previously been an English grocer's called Gallon and Son.
Five years later, the family - which also included Ram's sons Tajinder, Narinder and Rajinder - began to run a takeaway, the first in both Leeds and Yorkshire, although there were some sit-down Indian restaurants in existence.
In 1974, the Roundhay Road and Harehills area was still predominantly white British, with only a small Asian community and no other international food shops.
Ram's Indian sweets proved so popular with local residents that they began to ask for hot dishes too. Bobby's soon become a favourite of staff from nearby St James's Hospital.
After his sons and their wives took over the cooking, Ram would still visit the shop daily while well into his eighties.
As younger generations of the family had decided not to take on the business, the shop building has now been leased out.
Tributes to Ram and his sons have been pouring in on Facebook after the closure was announced, with many customers ruing the loss of the family's pakoras, chutneys and samosas.
Gerry Staniforth wrote: "Such a shame, a lovely gentleman owned this place. We have been going for years, probably 30!"
Andrea Dolan-Lyon added: "Love Bobby's. They supplied many a Boxing Day buffet, as well as work treats. So sad they have closed. Staff were fabulous and so friendly."
Mike Grossbard commented: "First curry I ever had was in 1977, a keema bhuna from Bobby's."
Bobby's last day of trading is today - Saturday July 22.