The agency has previously worked on the two events, which were brought under the Winter Festivals banner after a council funding review last month.
The deal follows the cancellation of Edinburgh's Hogmanay celebrations on New Year's Eve for the first time in its 11-year history. Health and safety concerns caused by severe weather conditions were to blame. Events affected by the decision included the Royal Bank Street Party, Seven Hill fireworks display and the Concert in the Gardens.
Edinburgh's Hogmanay costs about £1.8m to stage and the council estimates it has a £130,000 shortfall.
The cost of cancellation has been estimated at £220,000. This includes the refund of 8,000 tickets for the Concert in the Gardens.
A council spokesperson said: "Adverse weather cover would have put the insurance premiums too high so the council agreed with Unique Events that it wasn't required."
Last year's Hogmanay events generated £42m for Edinburgh.
Councillor Steve Cardownie said: "Although the 2003/2004 cancellation was a setback this will not deter us from ensuring Edinburgh's Winter Festivals go from strength to strength. Unique Events has an excellent reputation for cuttting-edge events and we look forward to working with them."
- Capital Entertainment's plans for New Year's Eve celebrations at the Millennium Dome were cancelled on December 19 because of the threat of a tube strike that never took place.