The 12-1 shot charged through on the inside rail in the last furlong of the two and a half mile (4,023 metres) Group One race to edge out Kingfisher (5-1) and 5-2 favourite Forgotten Rules.
Frankie Dettori's mount Forever Now (25-1) made most of the running until he was overhauled by the previously unbeaten Forgotten Rules with two furlongs to go.
"He gave me a great ride. I got on him in the parade ring and he was so switched off and relaxed and he raced that way," Lee told Channel 4 television after a victory that was worth 229,853 pounds ($365,374) to four-year-old Trip To Paris' connections.
"He was just conserving his energy and ... then there was a gap down the rails. I've had a good day in the office so I'll enjoy it."
Dunlop emulated the feat achieved by his father John who trained 1974 Gold Cup winner Ragstone.
"It's amazing," he said. "My parents have the Gold Cup in the dining room at home and it's been there since 1974.
"He's a very good horse now. He's won a Chester Cup and a Gold Cup which is a pretty big deal."
(Writing by Tony Jimenez, editing by Pritha Sarkar)
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