Johnson, 31, carded 10 birdies in his opening 15 holes and leads compatriot Brendan Steele by a stroke with Day among a group of players on six under.
Luke Donald is the best-placed Briton at the Muirfield Village course in Ohio on five under.
Rory McIlroy is one under and drew attention for a changed putting grip.
The Northern Irishman won his first title of the year at the Irish Open less than two weeks ago but said his success at the K Club was not down to putting.
"I had 32 putts on Saturday and 31 putts on Sunday," said the world number three.
"Any other week, you're not going to be doing too well. I hit a lot of fairways and a lot of greens, and that's what made that tournament."
Focus was on McIlroy, Day and world number two Jordan Spieth going into the tournament as all three had won on their last outings.
Spieth ended on two under, while his playing partner McIlroy suffered a double-bogey on the par-three 15th to end the day seven shots off Johnson.
"I rolled it well today," said nine-time PGA Tour winner Johnson. "I've been working pretty hard on the putter, and I felt like it's finally starting to pay off."
American Dustin Johnson shot an eight-under-par 64 to end the opening round of the Memorial Tournament two shots ahead of world number one Jason Day.