Running in the colours of Simon Munir and Isaac Souede, the eight-year-old was giving Elliott a third win in the race from the last four runnings.
It was a dramatic finish, with British raider Black Corton hampered by a loose horse close home, although Borice finished particularly strongly on the near side.
Conceding weight all round, the Paul Nicholls-trained Black Corton had been given a superb ride by Bryony Frost, and was always in the front rank as Dan Skelton's Azzuri set the pace.
Leading over the last two, it looked like another famous success was on the cards for the ground-breaking rider, but Borice (9-1) and Luke Dempsey would not be denied, winning by three and a quarter lengths.
Snugsborough Benny was third as the 8-1 favourite, with Peregrine Run fourth.
Elliott said: "I was worried about the ground with this lad. I thought it was too dry for him and I was umming and ahhing about whether to run him, but Simon flew over from Nice this morning and Anthony Bromley came over, and it was brilliant, unbelievable.
"He was sent over to us by Simon and Isaac and the plan was to go for good, big staying chases. I fancied him in the Paddy Power last year, but I'd say he hadn't filled into himself after coming from France and maybe I ran him a bit soon.
"But that was great he won today, and he'll be a nice horse for all those staying chases.
"It's been a lucky race for us, and it's nice to win another Galway Plate. We thought he was (our leading contender) all along - he was paper favourite this morning, but obviously drifted a bit with the ground."
Looking to the future, Aintree could be on the agenda.
Elliott said: "We'll enjoy today and dream about it, but that was what he was sent to me for - to be an English National horse.
"We said if we were going to run him, we'd run him in a good staying chase. I wasn't going to run him in a 20-grand race, it would have to be a 100-grand race plus, and thankfully it worked out."
He added: "Luke gave him a super ride, and there's no stronger jockey at that weight (10st 2lb). He's been around the yard a long time, and I'm good friends with his father, Philip, and it was nice for them that he won it."
Nicholls - the last British trainer to win the Plate when Oslot scored in 2008 - was understandably proud of Black Corton in defeat.
He said: "It was a fantastic run under that weight (11st 10lb). Bryony said she got into a bit of trouble with a loose horse, but that's racing.
"We were asking a lot of him with that weight, but he's not easy to place. It's great to come here and have a really good run.
"I honestly thought he had a monumental task and thought we'd asked him the impossible, but he nearly pulled it off."
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