The eight-year-old 5-1 shot led from the start in what turned into a two-horse race, with the rest well beaten on soft ground.
The pair were neck and neck as they leaped the final jump before Native River, who came in third last year, won a sprint to the finish and ultimately triumphed by four and a half lengths.
Anibale Fly (33-1) finished third in the flagship race of the four-day meeting.
"I’m speechless, he’s been a fantastic horse for me and I was lucky to pick up the ride on him. He's just amazing," said Johnson, who won the Gold Cup with Looks Like Trouble in 2000.
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"He jumps for fun. I was getting squeezed, but at the second-last and the last he just wanted it so much. Eighteen years feels a long time ago now."
Johnson was later fined 6,550 pounds by stewards and handed a seven-day ban for misuse of the whip.
Tizzard told ITV sport it was an unreal feeling to see Native River become the first to win at the second attempt.
"Watching the Irish winning everything for the last three days, we thought we'd got no chance," he said.
"He (Native River) wasn't quite right after Cheltenham last year so we've stuck to the plan and it has come in. I never thought in my biggest dreams that I would win a Gold Cup."
Second place was a disappointment for Might Bite's trainer Nicky Henderson, who had been hoping to become the first to win the Gold Cup, Champion Hurdle and Champion Chase at one Cheltenham Festival.
"It was a great race, to see two horses jump the last like that after a tough steeplechase, and you realise 12 others can't even get into it," said Henderson.
"Someone's got to finish second and we've made it a race. I'm very proud of the horse."
Last year's winner Sizing John was absent after suffering a fractured pelvis. It was the third year in a row the champion has missed the race after Coneygree and Don Cossack did not defend their crowns.
(Reporting by Alan Baldwin, editing by Toby Davis and Ed Osmond)
