He was known for the prophetic interpretation of dreams and visions, and encouraging the church to engage in the practice.
More than a thousand people paid tribute to Mr Jackson on his Facebook page, giving their thanks and condolences.
In a statement, his wife Diane said: "On behalf of the entire Streams family, I want to thank everyone for the prayers and outpouring of encouragement we've received over the last year.
"After recovering miraculously from cancer surgery last summer and returning to the activities and ministry work he loved, we found out a few months ago that the pleurisy and inflammation John Paul was experiencing in his lungs could be a sign of something more serious.
"Despite the heartfelt prayers from those closest to him and many of you around the world, my husband, John Paul went to be with the Lord peacefully at our home this morning surrounded by family and loved ones."
Mr Jackson founded Streams Ministries International in 1993.
Pastor Tony Cooke, the Director of the Streams Training Centre England, knew John personally. He told Premier: "He was not impressed by his own amazing gifting or anybody else's amazing gifting. He was a man who realised that gifting was only as good as the character that under laid it.
"I do think that his whole take on Biblical dream interpretation will be seen in years to come as being a significant contribution. My own personal feeling is that in fifteen years' time this will as common as laying hands on people for healing.
"When I first met him my ministry - whatever it was - was very much in the doldrums. And I remember he just very kindly said to me: 'Tony, your understanding of how big God is, is just too small'.
"And that actually changed my whole ministry and my whole approach. He was an immensely generous person."
Listen to Pastor Tony Cooke speaking to Premier's Aaron James:
