Rahm played an almost flawless final round having trailled Koepka by two shots after the third round was completed earlier on Sunday.
An emotional Rahm was greeted by his family once he sunk the final putt on the 18th hole as well as former Masters winner Jose Maria Olazabal.
He made four birdies and just one bogey in a round of 69 whereas Koepka faltered badly after leading overnight, and finished in a round of 75 to finish tied second with Phil Mickleson (52), who himself rocketed up the leaderboard with an historic round at Augusta - a seven-under 65.
Rahm adds to his 2021 US Open major and becomes the fourth Spaniard to win The Masters after Seve Ballasteros, Olazabal and Sergio Garcia.
After being presented with his green jacket by Scottie Scheffler (26) Rahm said: "The history of the game is a big part of why I play.
"For me to get it done on his (Ballasteros') 40th anniversary of his win and on his birthday was really special.
"I thought eight was a key birdie. I had to get through Amen Corner by par, so making sure I didn't put myself into a difficult position on 10, 11 and 12 was key."
With the victory Rahm also reclaims the number one world ranking from defending champion Scheffler.
Sam Bennett, who was part of the final group during the third round, won the silver jug as low amateur and called his walk up the 18th with the title secured "the coolest moment of his life".
