The Dane, who won this title in 2000 and 2002, hit nine birdies to finish the day on 14 under, just one shot behind leaders Alex Noren, Ernie Els and Alexander Levy.
The Swede had moved into a share of the lead thanks to five birdies in a steady round but a dropped shot on 15 saw him drop behind Bjorn.
An eagle on the next, however, leapfrogged him over the former Ryder Cup player and into the lead.
Els fired a 69 for the second consecutive round as he reached the turn in level par but three birdies on the back-nine, including one on the last, moved him back to the top of the leaderboard.
Levy also birdied the last as a 68 gave the Frenchman a share of the lead
Bjorn has finished second and in a tie for fourth in his last two events and is now hoping he can claim his first win of the season on Sunday.
"I played nicely the first two rounds," he told www.europeantour.com.
"I played really well and I needed that round to give myself a chance, I knew that when I set out today [Saturday] that it was important to find a round in the mid-60s to give myself a chance for tomorrow [Sunday].
"I've always played this golf course well. It suits my eye well and I would generally say I've got a pretty good wedge game and that comes in handy on this golf course."
Englishman Matthew Baldwin was sat alongside Bjorn after he mimicked Els with a second consecutive 69.
Play started an hour late due to the weather and around an hour and 20 minutes was lost in the early afternoon as the rain continued to come down - but that did not seem to distract the field too much as the men at the top continued to post low scores.
American Peter Uihlein followed up Friday's 66 with a 67 to sit just two shots off the lead as he recorded a single bogey on the 17th.
South Africa's Brandon Stone and Dutchman Joost Luiten both shot rounds of 67 to move to 12 under just three shots off the lead.
Home favourites Marcel Siem and Martin Kaymer were also on 12 under along with Danny Willett, who will be aiming to be the first man to successfully defend this title on Sunday.
Sergio Garcia was the only other player to shoot a seven-under par round to move four off the lead with seven birdies in a flawless 65.