Three ahead of the chasing pack overnight, the 26-year-old American ground out a two-under-par 68 on a fast-running layout in tricky, gusting winds to end the third round with a slender one-shot lead at 13-under 197.
Bradley came desperately close to finding water off the tee at the last, pulling his drive well left behind a large rock from where he had to chip out sideways on to the fairway before signing off with a five.
In pursuit of a fourth triumph on the PGA Tour, he mixed five birdies with three bogeys in hot, humid and breezy conditions to retain his place atop the leaderboard at the TPC Four Seasons Resort Las Colinas.
"I've been playing really well and obviously I like where I am sitting," Bradley, winner here in 2011 before going on to claim his first major title at the PGA Championship just three months later, told reporters.
"One of the most solid rounds of the year considering all that was going on. Today was obviously a lot tougher. In the early part of the round I was at one over par, hit some bad shots.
"But I really settled in and played really solid from there on out. I made some putts when I needed to so I am excited for tomorrow."
South Korean Bae Sang-moon was alone in second after firing a 66, with American journeyman Tom Gillis (67) a further stroke back at 11 under, having holed out from a greenside bunker to birdie the last.
Former Masters champion Charl Schwartzel of South Africa, playing in the Byron Nelson Championship for the first time, was among a group of four players knotted at eight under after shooting a 69.
TOUGH EARLY GOING
Bradley, aiming to become the tournament's first wire-to-wire winner since Tom Watson in 1980, initially found the going tough on Saturday, following a birdie at the third with bogeys at the fifth and sixth for his lead to be cut to one.
However, he rebounded with a two-putt birdie at the par-five seventh, then rolled in a 14-footer to birdie the ninth and reach the turn two strokes in front at 12 under.
Despite strengthening winds on a layout becoming increasingly fast and firm, the long-hitting American drained a 35-foot birdie putt at the par-four 11th to stretch his lead to three shots.
Bae cut the deficit to two when he sank an eight-foot birdie putt at the 13th and was poised to get even closer after hitting a superb wedge approach to just three feet at the par-five 16th.
Though the Korean went on to birdie that hole, Bradley matched him by splashing out of a greenside bunker and coolly rolling in a 12-foot putt.
Bradley did well to save par at the 198-yard 17th, hitting a stunning second shot out of a greenside bunker to three feet, but once again he struggled at his nemesis hole, the challenging 18th, which he had bogeyed in each of the two previous rounds.
"I thought it was in the water," the slender American said of his drive at the last. "But another foot to the right and I'm actually almost in the fairway, so it could have gone either way."
Bradley clinched his first PGA Tour title by beating compatriot Ryan Palmer in a playoff for the 2011 Byron Nelson Championship and said he would feed off memories of that breakthrough victory in Sunday's final round.
"A lot of stuff I'll draw upon," added the American, who took control of the tournament after opening with a blistering course record 60. "I know this course pretty well, I putt these greens very well. I look forward to a low round tomorrow."
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