Four-times champion Woods birdied three of the four par-fives in surprisingly firm and fast-running conditions at a sun-drenched Muirfield Village Golf Club to card a two-under-par 70, finishing four strokes behind Stallings.
Americans Spencer Levin, who totalled only 21 putts, and Erik Compton opened with 67s while Australian Aaron Baddeley, South African Rory Sabbatini and Argentina's Andres Romero were among a group of seven players knotted on 69.
Woods, who won the most recent of his four titles here in 2009, was reasonably satisfied after mixing four birdies with a double-bogey at the treacherous par-four 18th, his ninth hole of the day.
"It was just a solid round today," the 36-year-old said. "It certainly could have been a lot lower, but I'm pleased with the way I hit the golf ball today. I didn't do anything great and I didn't do anything poorly today.
"The golf course was playing quick and the fairways were starting to chase a little bit. The greens were somewhat receptive. It's a hell of a test out there, really."
Stallings, who has missed the cut in his last six US PGA Tour events while struggling to cope with a lingering rib injury, rocketed to the top of the leaderboard with three birdies in his last four holes.
"It was a great start," said the 27-year-old American who won his only title on the U.S. circuit in his rookie season at the 2011 Greenbrier Classic.
"I definitely haven't played very good up to this point. Lacklustre would be an understatement.
"But I stayed positive all throughout being injured and stuff like that. I kept telling myself that it was only a matter of time before a round like this was coming around."
British world number one Luke Donald and second-ranked Northern Ireland Rory McIlroy were relatively pleased with their scores, though both had to negotiate adversity before carding matching 71s.
"Hopefully that's my loose round out of the way," said Donald, who recovered from a double-bogey at the 18th, his ninth hole of the day.
"I didn't hit the ball well and I was scrambling a lot. But fortunately I putted pretty well, and I'm pretty happy with the score in the end. This course is tricky.
"There are some tougher pins out there today and overall I'll take one under and try and play better the next three days," added the Englishman who won the European Tour's flagship PGA Championship at Wentworth on Sunday.
McIlroy faced a much bigger hill to climb after he ran up an ugly quadruple-bogey seven at the par-three 12th, his third hole of the round.
"It wasn't the start that I wanted to get off to, being four over through three holes, especially after the last few weeks," said the 23-year-old, who has missed the cut in his last two tournaments.
"I was just like, 'Here we go again'. But I hung in there well, and proud of myself for the way I just fought back. To finish the round under par I thought was a really good effort."
Steve Stricker, who won last year's Memorial tournament, bogeyed four of his last six holes to open with a 73 while fellow American and Masters champion Bubba Watson carded a 75.
Four-times Major champion Phil Mickelson withdrew from the tournament after shooting a 79, citing fatigue.