Raphael Jacquelin of France finished with two birdies and an eagle for a 2-under 70. He is tied for second place with Jonathan Lomas, who shot an even-par 72, at minus-4.
Andrew Coltart snuck up the leaderboard with a 1-under 71. He is knotted in fourth place with Graeme Storm (74) at 2-under-par 214.
Bjorn began the third round tied for the lead with Jamie Donaldson, but found himself alone in first after strong wind knocked several players down the leaderboard. He parred his first three holes, then two-putted from 45 feet for birdie at the par-5 fourth.
Bjorn's lead reached three shots when he drained a 15-foot birdie putt at the ninth. He saved a nice par from the left rough at the par-5 10th and extended his margin with a birdie at the par-5 13th.
The Dane birdied the next par-5, No. 16, to build a five-shot edge. He found trouble at the 17th when he tallied his first bogey since the 15th hole on Friday.
Bjorn still had the par-5 closing hole to try and extend his lead, which thanks to Jacquelin's spectacular finish, had dipped to three. Bjorn had 3 feet for a birdie at the last, but missed.
His lead is still a comfortable four shots and the 34-year-old knows that in the difficult conditions Saturday brought, any round under par was sensational.
'I knew I had to go out there and find a solid game and it was certainly solid today,' said the two-time Ryder Cupper. 'It was as good as you want to play in these conditions and I don't think you can expect much more than I achieved today so I am very pleased with that.'
Jacquelin was two-over on his round until the par-5 16th. He was short in a bunker after his second shot, but blasted out to a foot for a tap-in birdie. The Frenchman knocked a 6-iron to 15 feet to set up birdie at the par-3 17th.
Jacquelin rolled in a 12-foot eagle putt at the last to get within four.
'You just have to keep trying and keep trying to hit the fairways because that is the key here, especially today with the wind,' he said. 'I am very pleased with the finish and to be 4 under, so we will see what tomorrow brings. Thomas is out there in front but we have 18 holes tomorrow, but I have a chance there is no doubt about that and I will try my best, definitely.'
Lomas was also two-over on the back nine, but closed with two birdies in the last five holes.
'Obviously it was a tough day for everybody out there and it was tempting to take on shots but you couldn't afford to in the wind,' said Lomas. 'I've kept to my game plan and that's what I've tried to do every day.'
Darren Clarke (75), Kenneth Ferrie (70), Richard Green (70), Peter Hanson (69), Damien McGrane (74), Nick O'Hern (72) and Brett Rumford (75) are tied for sixth place at minus-1.
Reigning U.S. Open champion Michael Campbell managed a 1-under 71 and is part of a group tied for 18th at 1-over-par 217.
Defending champion and two-time U.S. Open winner Retief Goosen struggled to a 5-over 77 and shares 23rd at plus-2.
Donaldson shared the second-round lead with Bjorn, but fell out of the lead very quickly. He was 3 over in his first two holes, then ended up with a 10-over-par 82 in round three. Donaldson fell into a tie for 46th place at plus-5.