Australian Jason Day had the day's best score, jumping within four shots of the lead with a 63, while defending champion Bubba Watson (66) vaulted up the field with five successive birdies, only to bogey the final two holes and slip six behind.
World number one Brooks Koepka also shot 66 to trail Sucher by eight.
Sucher has not finished better than 20th in his stop-start 36-event career on the PGA Tour. The 32-year-old is making his fourth start of the season after finally recovering from ankle and knee injuries that sidelined for more than a year and required surgery in 2017.
"Last few years I had were a bit of a struggle," he said. "It seemed the harder I worked the more the left leg would hurt.
"So this is the first year I can remember in a long time I'm pain free and feeling really good."
He is playing on a medical extension, and a strong finish on Sunday would help his cause as he battles to retain his exempt status for next season.
Sucher said his ankle had given him problems since his days playing high school basketball.
"I messed everything up in my ankle, but it was starting to come up the whole left side," he said.
"My knee was giving me all kinds of trouble. I had a torn tendon in there, cartilage was damaged, and it was just all stress from repeatedly swinging and not being able to turn on the ankle, which was causing the knee to hurt and wearing it down."
It took 13 months after ankle surgery before he could resume swinging a golf club.
"The lady was right who did the surgery. She said, 'after six months you'll be running, but you're not going to be able to twist on it very much'.
"The last four or five months have been great. I feel awesome now."
Spieth and Mickelson both missed the two-under cut.
Spieth, the 2017 champion, carded a one-under 69 on Friday but it was not enough after a first round 73. Mickelson, who shot 67 in the first round, was seven over for his first eight holes and ended with a 76.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Peter Rutherford)
