Scheffler, who used a back-nine eagle to get his round back on track, carded a one-under-par 71 on a firm and fast Augusta National layout to reach seven under on the week and put himself in position to win a second Masters.
With a win, Scheffler would become the 18th player to win the Masters multiple times and the first to accomplish the feat since Bubba Watson in 2014.
"Being patient out there I think is really important," said Scheffler. "Especially on a day like today. It was a frustrating day to be playing this golf course. It was so challenging."
Morikawa, who is seeking the third leg of the career Grand Slam of golf's four majors, began his day with three consecutive birdies and was alone in second place after a three-under-par 69 that was one shot off the day's low round.
Max Homa (73), who along with LIV Golf's Bryson DeChambeau (75) held a share of the halfway lead with Scheffler, was a further shot back in third while Ludvig Aberg (70) of Sweden was three shots off the lead in his major championship debut.
DeChambeau, who struggled with his putting most of the day, drove it into the trees at the par-four 18th, pitched out to the fairway and then holed out from 77 yards for an unlikely birdie that put a spring in his step.
"I just figured it was easier than putting," joked DeChambeau.
Scheffler had dropped out of the lead after a shaky start to the back nine where he followed a double-bogey at the par-four 10th with a bogey at the 11th.
But, unflustered, he joined Morikawa atop the leaderboard with a 31-foot eagle putt at the par-five 13th where his approach shot looked ready to settle some 10 feet from the hole before it suddenly gathered speed rolled away.
"I didn't know whether or not it was going to get there, and it kind of just nudged right over the edge and went in," said Scheffler. "So it was exciting, and it was nice to be able to steal a couple shots there on 13 and get back in the tournament."
Scheffler then picked up another stroke at the par-five 15th where he got up and down for birdie after his approach shot from 238 yards sped right off the back of the green and then offset his bogey at 17 with an eight-foot birdie at the last.
Denmark's Nicolai Hojgaard, who like Aberg is seeking to become the first Masters rookie to win a Green Jacket since 1979, was leading after three birdies around the turn but then made five consecutive bogeys to slide down the leaderboard.
Struggling Woods
Woods, fresh off his Masters record-setting 24th made cut, began his third round seven shots behind the leaders and hoping his injury-ravaged body would hold up and allow him to move into contention.
But Woods looked worn out from playing 23 holes on Friday and suffered an unwanted personal milestone with a 10-over-par 82 that tied the day's highest round.
"Just hit the ball in all the places that I know I shouldn't hit it," said Woods. "And I missed a lot of putts. Easy, makeable putts."
Following a three-putt bogey at the par-three fourth, Woods drained a fist pump-inducing birdie putt from 19 feet at the fifth before he suddenly started to unravel.
Woods bogeyed the par-three sixth after his tee shot landed behind the green, made a double-bogey at the seventh where he chipped into a bunker, carded another double at the eighth and a bogey at nine where his approach found another sand trap.
The 15-times major winner then mixed five bogeys with a lone birdie on the closing stretch.
Rory McIlroy, who began the day 10 shots off the lead and seeking a low score to boost his hopes of completing the career Grand Slam of golf's four majors, carded a one-under-par 71.
"All I can do is come here and try my best. That's what I do every time I show up. Some years it's better than others," said world number two McIlroy. "I've just got to keep showing up and try to do the right thing."
