In a battle royale approaching the line, Tiger Roll just held off a late resurgence from the Mullins' horse to edge a photo.
Pleasant Company had headed the race for most of the four plus miles but over the last fence, Tiger Roll took the lead and pulled away at the elbow. Tiger Roll is a yard favourite.
It is a second win in the race for Elliott, who guided Silver Birch to victory in 2007, while he also trained the third-placed finisher Bless The Wings (40-1).
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There have been 19 goals scored in the five previous Premier League meetings between these sides, at an average of 3.8 per game. And Firmino got in on the act in the final stages as Klopp's side move within a point of United, who have two games in hand.
"It's time to put the horse first rather than economics if we don't want to see dead horses".
In an attritional affair that saw numerous 40-runner field fail to complete, David Mullins led for a long way on his father Willie on Pleasant Company, who jumped superbly the whole way around the four-mile-plus course but couldn't hold off the fast-finishing Tiger Roll after he loomed upsides him before two out.
The eight-year-old victor, who has won three times at the Cheltenham Festival in recent years, including in the cross-country race at last month's event, had come into the race well-fancied. You watch it on television then you get a chance to ride in it. It's two years since I had an experience like that. I was actually fighting with Michael O'Leary, saying that when he retires he's to keep him at home at Gigginstown, and he said, no, he's not good enough, so he'd better keep him now!