Johnson narrowly missed sharing the top spot when his 12-foot birdie putt fell inches short at the 18th hole.
Dufner, seeking his third US Tour victory in his last five tournaments, used six birdies to offset bogeys at 14 and 16 for a four-under 66 and 195 total.
"If I were to pull the victory off tomorrow it would be pretty great company to be in," Dufner said of the possibility of joining Ben Hogan as the only two golfers to win the two Dallas area tournaments in the same year.
Johnson, who carded a five-under 65, had other ideas.
"I hope he's getting a little tired of winning," the 2007 Masters champion said.
He pressured Dufner throughout the hot day, using a birdie at the par-three 16th that Dufner bogeyed, to move within one.
Dufner steadied himself to join Johnson in a birdie-par finish as they pulled away from the field.
"Hopefully both of us can go out there tomorrow and increase that distance from them and make it kind of a two-horse race ... and give the fans and the people watching at home some good golf and some good theater," said Dufner.
Americans Tom Gillis (203), Bo Van Pelt (204), John Huh (205) and Ryan Palmer (205) rounded out the top six.
Dufner had never won a US Tour event until he captured the New Orleans Classic on April 29. He took the next week off to get married before tying for 68th at The Players.
Last week, he was back on course, claiming the Byron Nelson Championship at Irving, Texas.
Dufner entered Saturday's play with a two-stroke lead before Johnson birdied four of the first nine holes.
"Yes, I got off to a good start," Johnson said. "Today I just scored. I think Dufner played better than I did. But I scored."
The early spurt by Johnson seemed to ignite Dufner as he rattled off birdies at 10, 11 and 12 before his streak of 38 consecutive bogey-free holes was snapped at the 449-yard 14th when he wound up in the rough, then missed the green with the second shot.
But Johnson three-putted for a bogey.
The two parred the 15th, before Johnson took advantage of a two-stroke swing at the par-three 16th to close within one when Dufner missed a par putt from seven feet while Johnson sank a 16-foot birdie putt.