Scheffler also extended his streak to 49 holes without a bogey dating to the third round of The Players Championship two weeks ago, where he became the first back-to-back winner in the 50 years of the PGA Tour's flagship event.
The streak that matters is his bid for a third straight PGA Tour victory, a feat last accomplished seven years ago by Dustin Johnson.
That also was the last thing on Scheffler's mind at Memorial Park on Thursday, his final tournament before he heads to the Masters as a strong favourite.
"I don't ever really put expectations on myself," Scheffler said. "I try to be committed to my shots. I try to stay patient out there. At the beginning of the week, I'm not looking forward toward Sunday. I'm just focused on today."
Moore came up short of the first hole and missed a 6-foot par putt. That was his lone mistake. He pitched in for eagle on the par-5 third and was on his way, getting up-and-down on the 18th for a 64.
"After the first hole, just tried to see how many greens I could hit," Moore said. "Got off to a little bit of a jump start there on 3, chipped in for eagle and birdied 4. Just got into the round."
Furr matched him late in the day, holing out from 120 yards for eagle on the par-4 fifth and closing with birdies on the par-5 eighth and par-3 ninth - making a 20-footer on 9.
"Had a good number. it was downwind and it landed right behind it, ripped right back to the hole and just never looked like it was going to miss," Furr said about the eagle.
The 25-year-old American is in his first season on the PGA Tour after earning a card last year on the Korn Ferry Tour.
"It's a dream come true, for sure," Furr said. "Just being surrounded by the best golfers in the world, it's kind of inspiring day to day. Can't ask for a better job than that."
Scheffler was a stroke back with Joe Highsmith and Davis Riley.
Canadians Adam Svensson and Mackenzie Hughes were at 66 with Alejandro Tosti, Aaron Rai and Beau Hossler. Peter Malnati, coming off a victory last week at the Valspar Championship in Florida, shot 68.
World No.1 Scheffler decided to take a little more time off last week at home in Dallas, mentally spent from winning the Arnold Palmer Invitational and The Players in consecutive weeks.
He had a rugged start at Memorial Park, having to make a 10-foot par putt on No.10 and not having his first birdie putt until his fourth hole at No.13. He made a 12-footer.
Cam Davis (68) and Harryson Endycott (69) are the leading Australian contenders.
Davis, who is tied 17th, had an up-and-down round, mixing six birdies with four bogeys.
The highlight of Endycott's round was a brilliant eagle on the 609-yard eighth, when he reached the green in two and then sank his 33-yard putt.
