Even with two bogeys the last three holes, Palmer finished a stroke ahead of playing partner Sergio Garcia after the two started the day tied for the lead.
Garcia shot a 74 while also making bogeys at the 16th and 18th holes, where he missed makeable putts.
“It was the kind of day if you let yourself go, it could go very wrong,” said Garcia, the 2004 Nelson champion who hasn’t won since 2008. “We grinded it out the best we could.”
The last three holes at TPC Four Seasons played into the wind, sustained all day at 25 mph with gusts near 40.
Garcia missed a 5-foot par putt on the 535-yard 16th hole, where Palmer also bogeyed after missing the fairway with his tee shot.
Palmer, who is letting caddie James Edmondson call the shots this week, missed the fairway again at No. 18 and two-putted from 27 feet once he got on the green. Garcia hit his drive way right and still had a chance to convert an 8 1/2-foot par saver that would have put him back in a share for the lead.
Only eight of 74 players shot under par Saturday. The best was a 67 by Arjun Atwal that tied him with Ryuji Imanda (70) for third place at 3 under.
“Obviously, the scores show how hard it was,” Palmer said.
Gary Woodland and Matt Kuchar shot 68s and were tied with Joe Ogilvie (72) for fifth at 2 under.
Local amateur Jordan Spieth opened with consecutive birdies before going on to a 72. He was among eight players tied for eighth place, only four strokes behind Palmer.
Spieth’s round ended about 4 p.m. local time, the same time the other 245 boys in his senior class at Dallas Jesuit Prep were beginning their graduation ceremony on the SMU campus about 20 miles away. Spieth didn’t try to make it to the ceremony, also near where Cowboys quarterback Tony Romo was getting married.
Palmer had a bogey at No. 6 where his first two shots found the rough before a two-putt from 19 feet, but got that stroke back with his only birdie, a 9-footer at No. 10.
“The front nine, I was actually kind of cruising along, not letting the wind bother me,” Palmer said. “It got tougher and tougher. … The last five holes are tough enough on a calm day. I don’t feel like I played that bad. It was a hard day of golf.”
Before this week, Palmer had made the cut only once in his seven previous appearances at the Nelson with nine consecutive rounds over par. Now he’s going into the final round at Lord Byron’s tournament with the lead.
“If you would have told me Thursday I’d have a one-shot lead, I would have called you a liar,” Palmer said. “We didn’t expect to be in this position at the beginning of the week.”