Woods carded four birdies and two bogeys in an opening 70 to lie four shots off leaders Brooks Koepka and Bryson DeChambeau.
Seeking a fifth Masters title but a first since 2005, Woods had quietly reached the 14th hole in two under par before a pulled tee shot set up a moment of magic from the former world number one.
Spotting a gap high in the pines, Woods threaded his second shot perfectly through the opening and on to the green, where he holed from 25 feet for an unlikely birdie.
That took the 43-year-old into a share of the lead but he bogeyed the 17th and had to save par on the last after finding sand off the tee.
"I though I played well today. I thought I hit a lot of good shots and if I missed I missed in the correct spots and had some of the simpler up and downs because of that," Woods told ESPN.
"I missed a few (putts) for sure, misread a couple and hit one bad one at six but other than that a good solid day."
I tried to take care of par fives when I could. I tried to stay mistake free, but that's very hard to do here. I tried to avoid bogeys and I only made two today, so it is a good positive start.
"I feel good and feel I can move on from this. When I've won here before I've shot 70 in the opening round, so I'm off to a good start
Coincidentally, Woods shot 70 in the first round of each of his first three Masters victories and since lying 33rd after an opening 74 in his 2005 triumph, the last 13 champions were all inside the top 10 after the opening round.
