After losing the first set to No. 2 ranked Murray, Djokovic rallied to win 3-6, 6-1, 6-2, 6-4 for his 12th major to draw even with Roy Emerson for fourth place on the all-time title list.
"It's really a very special moment," Djokovic said, "Perhaps the greatest moment of my career. I felt today something that I never felt before at Roland Garros, I felt the love of the crowd."
The 29-year-old has won the past four Grand Slam finals to become the third man to hold all four major crowns at the same time, joining Don Budge (1938) and Rod Laver (1962, 1969).
Since losing the 2015 final in Paris, Djokovic has won 28 Grand Slam matches in a row.
Previously a three-time finalist in Paris (2012, 2014-15), Djokovic captured the lone piece of missing major trophies.
With 12 majors, Djokovic is only behind Federer (17) , Nadal (14) and Pete Sampras (14) in Grand Slams.
When his victory was over, Djokovic took a racket to draw a heart in the red clay and dropped down on his back.
Djokovic drew the heart to mirror Gustavo Kuerten's celebration when the Brazilian won the French Open in 2001. Kuerten looked on, seemingly approving.
"I drew the heart on the court, like Guga, which he gave me permission to do," Djokovic said. "My heart will always be with you on this court."
It was another close result for Murray, who has lost eight out of 10 meetings at the Grand Slams against Djokovic, including the last three finals.
"This is his day today," Murray of Scotland said on court of Djokovic, his voice cracking. "What he has achieved the last 12 months is phenomenal. Winning all four of the Grand Slams in one year is an amazing achievement. This is something that is so rare in tennis, it hasn't happened in an extremely long time. And it's not going to happen for a long time."