Last month Harrington won the Cheltenham Gold Cup with Sizing John.
"These are all firsts for me - it's not bad for a 70-year-old," said Harrington, whose horse, ridden by Robbie Power, won at 9-2.
Gordon Elliott's Bless The Wings was runner-up for the second year in a row while Abolitionist took third.
The winner is owned by the Cooper Family Syndicate which comprises Sloane, Nigel and Billy Cooper, their sister Hazel and her husband.
"They bred him, they own him, they refused a lot of money for him this year and they are fantastic," said Harrington.
Thirteen of the 28 runners came from Ryanair boss Michael O'Leary's Gigginstown House stables, with the best-placed being the 2015 winner Thunder and Roses in fourth.
Despite running in his first handicap and for just the fourth time over fences, Our Duke never looked in any real danger and has now been cut from 25-1 to 14-1 for next year's Cheltenham Gold Cup.
"This means so much to me, it's my local racecourse and I was second in it a few years ago so this is nice compensation," said 35-year-old Power, who was on board Sizing John at Cheltenham and rode the 2007 Grand National winner Silver Birch.
"Once he got a good view of his fences and got plenty of light we were going to be happy.
"He just travelled so well everywhere, he is just a very good horse."
As well as three Cheltenham winners, Power also won four races at Aintree's Grand National meeting.
Trainer Jessica Harrington celebrated her first success in the Irish Grand National when favourite Our Duke won Monday's big race at Fairyhouse.