Karlsson finished at 6-under 207, despite a quick turnaround from his home in Monaco after he thought he'd missed the cut. It's the lowest round since the European PGA event was first played on the course in 1955. He's tied for second with Danny Willett (70).
Karlsson flew home to Monaco on Friday before receiving a phone call saying he was playing on the weekend. He flew to Paris, slept for two hours, then paid $11,000 for a private jet to take him to London, arriving at the course an hour before tee time.
He had nine birdies on Wentworth's redesigned West Course.
"Don't touch my putter, it might burn you," Karlsson said. "Luckily, I snatched a couple of hours sleep at Orly (Airport in Paris). After making three early birdies, I just decided to keep going.
"When you start like that it is easy to sit back and protect those birdies and maybe shoot a 70, but I decided to keep going. My caddy said at the 9th: 'We've been invited to this party, so let's party.'"
The 22-year-old Wood has top-five finishes at the last two British Opens, and is looking for his first European Tour victory.
"It's good to be leading," Wood said after holing three straight birdies from the second green. "I went off like a train after making a couple of nice putts and I stayed calm."
Wood drove into trees to bogey the 13th and 15th holes but quickly regained his advantage with a birdie at the 16th.
Willett failed to find consistency, mixing seven birdies with four bogeys and a double.
Overnight leader Luke Donald dropped back to fourth place and 5 under after a 72, and five players were left tied in fifth place at 3 under. They included defending champion Paul Casey of England, who shot a 72.
Ernie Els had a 76 to trail the leader by 10 strokes. Lee Westwood (73) was two more strokes behind