Slow, left-arm spinner Kline, who took part in 13 Tests between 1957 and 1961, played 88 first-class matches in which he took 276 wickets at 27.39. He died on Friday, Australia's cricketing authority said.
In the Brisbane Test of the famous 1960-61 series against the West Indies, Australia needed just one run to win when Kline fended off a delivery to square leg and his batting partner Ian Meckiff scrambled for the single.
"Throughout his cricket career he was involved in some extraordinary moments that have become part of the rich history of our great game," Cricket Australia Chief Executive James Sutherland said in a statement.
"It has been a very sad year for Australian cricket with the passing of a number of treasured members of our community including Richie Benaud, Arthur Morris and now Lindsay Kline.
