The German-born musician’s manager said he died at his home “peacefully and in the presence of his family”.
Last sold millions of records with his trademark “happy music” and was a regular fixture on British TV for many years.
He appeared at the Royal Albert Hall in the spring as part of a farewell tour he announced after becoming seriously ill last year.
The illness, which took a “life-threatening” turn last September, apparently forced him to face the fact that “a man full of plans, needs to not just slow down but give up his life on tour altogether”.
But he said the tour would give him the opportunity to bid farewell to his fans.
“The main thing is that my fans have the best concerts of their lives and we will make this our ‘happiest’ concert yet,” he said when he announced the tour in February.
Last started his music career in the 1960s, making instrumental recordings as James Last and His Orchestra – a big band with additional strings and a choir.
Over his career he produced 75 hit albums in Britain alone, and at his peak, was creating two albums a month.
The BBC spoke to the musician in March as he performed his final British shows – his 89th and 90th concerts at London’s Royal Albert Hall.
“I can’t talk about goodbye – it’s a terrible feeling,” he said.