The 36-year-old, who had finished in a tie for second behind Woods in the 2019 event, went one better after producing a brilliant display over four days at Augusta National.
He set a new record aggregate for the event with a 20-under-par 268 total in claiming a five-shot victory over Australian Cameron Smith and Korea's Sungae Im.
Johnson, who had his brother, Austin, caddying for him in the rescheduled major following its postponement in April due to COVID-19, is the first world No 1 to win the Masters since Woods achieved the feat in 2002.
"I have been dreaming of putting that Green Jacket on my whole life," Johnson told Bubba Watson, a two-time champion, as he made his way to the recorder's area after a celebratory hug with his fiancee, Paulina Gretzky.
Shortly afterwards in the Butler Cabin, he added: "As a kid, dreaming about winning the Masters, having Tiger put the Green jacket on you, it still seems like it's a dream. But I'm here, and what a great feeling it is. I couldn't be more excited."
Johnson, who was born less than 80 miles from Augusta in Columbia, South Carolina, had started the final round with a four-shot lead in the final men's major of 2020.
That advantage had been cut to just one after he stumbled to back-to-back bogeys at the fourth and fifth before a timely birdie-2 at the sixth set him on his way to a resounding victory.
"It's always tough to get it done in a major, no matter how good you're playing," said Johnson, who had let winning 54-hole positions slip in four previous majors.
"I was nervous all day. I could feel it. The Masters to me is the biggest tournament, it's the one that I wanted to win the most, and I felt it all day. I'm very proud of the way I handled myself and the way I finished off the golf tournament."
On having his brother at his side, he added: "It's an unbelievable feeling to experience all this with my brother. "I love him being on the bag. I had to jab him a little bit on 18 because he was tearing up and made me tear up. But it was a lot of fun. I'll remember this for the rest of my life and having Paulina there today, too, was awesome."
Johnson, who had come into this event on the back of four successive top-10 finishes at Augusta National and also a first FedEx Cup win on the PGA Tour in September, trimmed two shots off the record winning total set by Tiger Woods (1997) and Jordan Spieth (2015).
"It's an incredible feeling," he said of that feat. "I played unbelievable golf all week. The conditions of the course definitely helped the scoring a little bit, but I still played really well today. I thought it played really difficult. The wind was very tricky.
"To have the scoring record, shooting 20 under this week, it's a great honour, and I'm just so excited that it's hard to even talk."