The three-year-old held off a late challenge from outsider Egyptband, with Volvoreta in third place at Longchamp on Sunday.
It was a stunning display by Sinndar, already victorious in the Vodafone Derby at Epsom and the Budweiser Irish Derby.
Trained by John Oxx and owned and bred by the Aga Khan, he is the first Irish-trained winner of the Arc since Alleged's second triumph for Vincent O'Brien in 1978.
Sinndar had belated assistance in the race from stablemate and pacemaker Raypour, who hit the front later than expected.
The second favourite took over in the straight and had enough of a cushion to hold off the fast-finishing Egyptband.
Meanwhile Montjeu, bidding for a second successive Arc victory, struggled to make headway from midfield.
Sinndar is the first horse to add the Arc to victories in the Derbys at Epsom and the Curragh.
His jubilant jockey Johnny Murtagh waved a small Irish flag as he entered the winner's enclosure to loud cheers.
Murtagh, who landed an Arc-day treble, said: "I feel on top of the world at the moment. This is one hell of a racehorse."
But there was disappointment for Montjeu trainer John Hammond, who struggled to explain his horse's poor run.
The Chantilly-based Briton said: "For one reason or another, he hasn't run to his best. He just didn't fire.
"It's very unusual for a horse as consistent as him to throw in a moderate run like that."
Ten horses had lined up for the richest event of the European flat racing season, boasting a purse of 10.5 million francs (about £1m). It was the smallest field for the race since 1946.
Sinndar's connections were given a boost before the Arc when Namid, also trained by Oxx and ridden by Murtagh, cruised home in the Prix de l'Abbaye.
The four-year-old won the Group One race in comfortable fashion, ahead of Superstar Leo, with Tim Easterby's Pipalong in third.
Murtagh completed his treble on Petrushka, overcoming the considerable disadvantage of being stuck in the number 14 box on the wide outside in the Prix de l'Opera-Hotel.
Frankie Dettori took the starring role at the beginning of the day as Kabool emerged victorious in the Prix du Rond-Point-Casino.
The Italian had his Godolphin mount in front from the start and just held Kingsalsa with the Yorkshire raider Sugarfoot, second in this Group Two race last year, third.
Prix de l'Arc de Triomphe, 1m 4f, Longchamp
Pari-Mutuel (Returns in francs, includes one franc stake)
1 Sinndar J Murtagh 2.50 (1.50 place)
2 Egyptband O Doleuze 3.80
3 Volvoreta O Peslier 2.20
Distances: 1 1/2l, 3l
10 ran