The 31-year-old American, who missed the cut in seven of his 11 prior majors, shot a three-under par 69 to finish 72 holes at Augusta National on one-over par 289, matching the highest winning score in Masters history.
"Being Easter, Jesus was with me every step of the way," Johnson said.
"I felt Him. It was awesome."
Woods shared second with a three-over 291 with South Africans Retief Goosen and Rory Sabbatini, while American Jerry Kelly and England's Justin Rose shared fifth on 292.
Australian Stuart Appleby, the one-shot leader over Woods and Rose going into the final round, shared seventh on 293 with Padraig Harrington, who was attempting to be the first Irish major winner in 60 years.
Johnson, whose only prior PGA victory came in 2004 at Atlanta, won $1.26 million from a total purse of $7.25 million after missing the cut in the 2005 Masters and sharing 32nd at Augusta last year.
"Zach has been a very good player for a long time," Goosen said of Johnson, who became the first Masters winner not to play in the final pairing since Nick Faldo in 1990.
"He hits the ball straight and low and hits it accurately and that's what you need to do around here this week."
In a week of gusty winds, lightning-fast greens and frigid conditions, Johnson's final total equaled the record worst score by a winner, with Sam Snead in 1954 and Jack Burke in 1956 also carding totals of 289 in their Masters wins.
"Under the circumstances, that's great golfing," Sabbatini said.
"That was a phenomenal round."
Johnson began his final round with a bogey but followed it with birdies at the par-5 second and par-4 third, before bogeying the fifth and birdying the eighth, but his championship run began as he exited Amen Corner, the famed 11th-13th holes at Augusta.
The Iowa-born golfer layed up at the par-5 13th to set up a 2.5-metre birdie putt that gave him the lead, and followed with a birdie at 14 to seize a two-stroke advantage over two-time US Open champion Goosen with Woods four strokes back.
Woods then miraculously eagled the 13th after rolling his second shot close to the pin, claiming a share of second, two behind Johnson.
The turning point came on the next hole when Woods just missed a birdie chance on 14, while Johnson sank a three-metre putt for birdie at the 16th to reach level par and lift his lead to three.
Woods found the water at the par-5 15th but salvaged par while Johnson made bogey at the 17th and was just short of a bunker at the 18th green before chipping inches from the cup and tapping in his par.