Ridden by Billy Egan, Garachico ($61) scored a half-length win from the $5.50 favourite Newfoundland with Whisky On The Hill ($6) the same margin away third.
Garachico came out from France at the end of 2022 in the new ownership as the winner of one race at Evry, and Tuesday's outing was his 15th in Australia.
"It's been a long time coming," Sutton said.
"He's been a promising stayer throughout his time here and we've just been waiting to get him out to these extended trips.
"He ran third in this race last year and it was nice to go two better today.
"We've always had this race sort of circled. When you get these horses that love these types of trips, there's limited options for them and it's a great race.
"He loves a big galloping track like Flemington, so it was always the plan."
Sutton said winning a race on Melbourne Cup Day for a young trainer was a huge thrill.
He went close to capturing a Group 1 race when second in the Caulfield Guineas last month, while that horse Feroce finished third in the Carbine Club Stakes on Victoria Derby Day.
"It's been a fast rise coming through the training ranks, and to be able to have this calibre of horse to bring to the races on days like today, it's just an unbelievable thrill," Sutton said.
"I feel very honoured to be able to be in this position.
"It's nice to do it with the young, progressive horses that are coming through, but you add another string to your bow by showing that you can train these European stayers, and then you get them to the point where they can deliver on the big stage over these extended trips."
