He dropped to 8 under after his third bogey of the day on No. 13, but reeled off three birdies over his final five holes, including one on the hard 18th hole to post a 4-under 68.
Woods missed a short birdie putt on the 14th and had three-putt bogeys on the 16th and 18th holes to finish in a tie for seventh at 9-under along with Whee Kim of South Korea (a 67 on Saturday), J.B. Holmes (71) and Japan's Hideki Matsuyama (71).
Niemann is contesting just his fourth PGA Tour event since turning pro immediately after last month's Masters. On a soft course yielding low scores, he made seven birdies to join Stanley (66) at 11-under 133.
And things looked good when he birdied the first hole, but Woods could never get closer than 3 strokes to the leaders and again missed numerous changes, settling for just three birdies and three bogeys.
Nicklaus however, who will host this week's Memorial tournament in OH, said he is convinced Woods can win Majors again after playing with him on Monday at Shinnecock Hills, the venue for the 2018 US Open. He was six behind.
Woods gained 14 strokes on the field from tee to green - the best in the field - as he threatened to win his first PGA Tour title since 2013 and sixth at Muirfield Village. The 26-year-old carded seven birdies to overpower two bogeys at hole 7 (3-putt from 25'5") and 17, posting 1.643 putts per GIR and 3.269 SG: "Putting with 28 total putts. "I played really, really well".
Former world No.1 Jason Day, a resident of nearby Columbus and a member at Muirfield Village, has dropped two shots to sink to four under.
"I haven't had great finishes here, so the expectation of me playing is a little bit different this year".
What to have in your hurricane kit this season
And as so many areas are still recovering from Harvey, Irma and Maria, get ready for above average hurricane season for 2018. The hurricane season is here and runs until November 30, normally peaking in August and September.
"The break was a factor as I missed my putt on 12 and after that I missed everything", he said. He played a marvelous flop shot out of deep rough to a couple of feet for a certain par, only for DeChambeau to hit his approach 12 feet behind the hole and make the birdie.
For the week, Woods made just one birdie over the closing three holes and played them in 3 over par.
Woods was second to last in the key putting statistic among the 73 players who went all four rounds. Woods remains hopeful saying, "I'm in position where if I shoot another good round like I have the last two days I've got a chance".
Getting off to a good start on an early tee time has its advantages.
Rory McIlroy didn't figure to be in contention at the Memorial, but he showed how much difference a stellar Saturday round can make. He played bogey-free for a 64 and wound up just six shots behind. Many of them lingered at the 18th green after spending much of the final round as if this might be the day Tiger Woods returned to winning. McIlroy missed three birdie chances inside 8 feet.
Since his closing 64 at the Masters, Spieth has finished at least 12 shots behind the victor in his three tournaments and missed the cut in his final event before heading to Shinnecock Hills for the U.S. Open.