The track, which went to number two in 1987, beat the original Band Aid single - recorded in 1984 - into second place.
Wham's Last Christmas, also released in 1984, polled at number three and Slade's Merry Xmas Everybody - first released in 1973 - was at number four.
Current chart-topper Do They Know It's Christmas? came in at number five.
The re-recorded charity single by Band Aid 20, widely tipped to be this year's Christmas number one, has already sold more than 600,000 since its release on 29 November.
Also in the top ten were Bing Crosby's White Christmas, at number eight, and John Lennon's Happy Xmas (War is Over) at number nine.
However Sir Cliff Richard failed to make the top 10, despite a string of Christmas hits including Millennium Prayer and Mistletoe and Wine.
The poll of 10,000 people was conducted online. Fairytale of New York gained nearly a quarter of all votes cast.
Singer Kirsty MacColl died four years ago at the age of 41 when she was hit by a speedboat while scuba diving off the coast of Mexico.