The 71-year-old veteran broadcaster and charity fundraiser, who was admitted to Killingbeck Hospital in Leeds, West Yorkshire, underwent a three-hour quadruple bypass operation on Thursday.
His secretary, Janet Cope, said: "He's talking to the surgeons and nurses, smiling and responding well to the operation."
Sir Jimmy learned he needed the heart operation four years ago but continually put it off, Mrs Cope said. "He kept going to check-ups and putting it off until a few weeks ago, when his sister Chrissie, who was 76, died from a heart attack.
"I think that made him realise it was time for him to go in to for the operation."
Sir Jimmy was expected to remain in hospital for two weeks and must then take it easy for three or four months.
His trademark cigars were not the reason for the operation, according to the doctors, said Mrs Cope. "They say it is nothing to do with it. He has a huge cigar burning all the time but he never inhales, its just a showbiz thing," she explained.
It is estimated he has helped raise a total of more than pounds 30 million for charity, including pounds 12 million to rebuild the National Spinal Injuries Centre at Stoke Mandeville, Buckinghamshire, where he partly lives and works