Price had twice finished as the runner-up in The Open, in 1982 and 1988, but this was his time.
After birdieing Turnberry's 16th, he made an incredible eagle on 17, courtesy of a mammoth putt across the green.
Parnevik bogeyed 18 up ahead, meaning a par at the last was enough for Price to secure victory.
"It was probably the most emotional time - outside of my kids being born - that I've ever had in my life," said the Champion.
"It's like a fairy tale to finish like that to win a Major Championship. You reach down deep inside yourself and, sometimes, you surprise yourself."