Moller Nielsen was admitted to hospital in Odense last June after becoming ill and, although he was initially discharged after a few days, later scans determined he required an operation on a brain tumour.
He underwent surgery the following month, but struggled to regain full health.
After spending much of last summer bedridden at his home in Kerteminde, he was re-admitted to hospital. Shortly before Christmas he was transferred to the Fyn Hospice, where he died early on Thursday morning.
Former Manchester United and Denmark goalkeeper Peter Schmeichel, a European champion 26 years ago under Moller Nielsen, led the tributes.
He posted on Twitter: "Richard Moller Nielsen, an inspiration, my teacher and my friend. R.I.P my friend."
Moller Nielsen coached numerous Danish clubs after ending his seven-year playing career with hometown club Odense.
After then serving his international apprenticeship as Under-21 coach and first-team assistant coach, he took over the senior Denmark hotseat for what turned out to be a six-year stint in 1990.
The undoubted highlight arrived in Sweden in 1992, when Denmark - a late replacement in the competition for the former Yugoslavia - were crowned champions thanks to a 2-0 victory in the final over favourites Germany.
He went on to also take charge of the Finland and Israel national teams before retiring after a short spell with Kolding FC in 2003.
Moller Nielsen is survived by his wife Jonna, children Birgitte, Christian and Tommy and six grandchildren.
His family expressed their gratitude on www.dbu.dk for all the messages of support received throughout Moller Nielsen's illness.