Marmelo claimed the lead inside the final furlong of the $7.3 million race but was grabbed late by the fast-finishing Cross Counter who won by a length, with a further two lengths to A Prince Of Arran in third.
Sheikh Mohammed has been trying to win the Melbourne Cup for 30 years.
Regal Monarch, trained by Chris Waller, was euthanised after falling in race four previous year, while British stayer Red Cadeaux was put down two weeks after falling and breaking a leg in the 2015 Melbourne Cup.
"Today's tragedy is a stark reminder of the risks these horses face every time they run, and highlights that behind the gloss and glamour of Cup celebrations there is a dark reality. And a third Melbourne Cup, woo!" said an ecstatic McEvoy, who also won in 2000 and 2016.
It was later revealed the horse had suffered a broken right shoulder and couldn't be saved.
Cross Counter wins the 2018 Melbourne Cup.
As the first British trainer to win, he said: "I'm so lucky.to be in this position to be able to train these horses".
Appleby has not had time to think about becoming the first English trainer of a Melbourne Cup victor, with his focus on the significance of the win for Sheikh Mohammed and Godolphin.
The Chairmen and Directors of the Victoria Racing Club (VRC) later extended their sympathies, with Flemington Racecourse tweeting their condolences, "To the Williams family and the connections of The CliffsofMoher, which was euthanised after he went amiss in the Melbourne Cup".
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She and her husband grabbed their dog, jumped in their pickup truck and drove through flames before getting to safety, she said. Local media reports have shown images of traffic moving away from the plumes of smoke and orange glow of the wildfire.
"That piece of work cemented that the horse was well on track", Appleby said.
"What a thrill, to do it for Sheikh Mohammed and Charlie Appleby".
"For me it's obviously very special to be here, but for the team, it's just fantastic".
"Charlie was travelling head lad when I was with Godolphin in the United Kingdom and we used to head up to all the Northern tracks a lot of the time". "I'm over the moon".
Jamie Stier, Racing Victoria's manager of integrity services, confirmed the news on Tuesday afternoon, The Age reported.
Past year it was reported his efforts to win the Cup, combined with other racing operations Down Under, had set him back $1.1 billion. "Melbourne see you next year", Fellowes, who is the chairman of Newmarket Open Weekend, wrote on Twitter after the race.
Ahead of the race, Aidan O'Brien-trained Yucatan, owned by six-time Melbourne Cup victor Lloyd Williams, was strong favorite but the sodden conditions threw the form book out the window.
In a field dominated by worldwide runners, Youngstar was the best-placed of the Australian-bred horses, the Chris Waller-trained stayer finishing sixth.