South African Richard Sterne was one under through four holes to join McDowell and France’s Romain Wattel, who like Hansen was yet to tee off.
Overnight leader Hansen has played just ten events so far this season and will undergo an operation on his left hand on Tuesday that will keep him out of several more, but insisted he was looking forward to it.
"My hand feels fine but I'm having my operation on Tuesday, so I'm looking forward to getting that out of the way and hopefully be ready for the end of the season," the two-time PGA Championship winner said.
"It's a problem I have had for years and years and years that I have managed with cortisone injections before.
"There's basically no meniscus in this part of the hand so it goes straight, bone on bone, so no cartilage. They are going in in three different places and they are going to remove a cyst I have here and clean this up and there's a 50/50 chance it's going to be okay.
"The other thing is to take the whole bone out and cut it a bit, but that's a big operation and I would be out for six months. So we are hoping to do that (the first option) and it's going to be good enough."
Björn, who has been fourth, second, second and 18th in his last four events, saved par after missing the green on the 17th, but was unable to repeat the feat from a greenside bunker on the 18th.
That dropped him back into a tie for the lead with Hansen and Sterne, who had birdied the tenth and 14th to improve to five under par.
McDowell remained four under while the third member of the group, France's Grégory Bourdy, was two under after an amazing birdie on the 18th. Bourdy's hooked tee shot was heading for the water before it hit a tree and bounced back into the middle of the fairway, from where he hit his approach to 25 feet and holed the putt.
Björn was briefly overtaken at the top of the leaderboard when England's James Morrison birdied the first, second, fifth and sixth to improve to six under par.
But Morrison then bogeyed the eighth as Björn was making a birdie four on the par five third which lifted him back into the outright lead once more.
Morrison was one of five players sharing second place, with Scotland's Stephen Gallacher, 2007 winner Graeme Storm and Spain's Alejandro Cañizares all on five under approaching the end of their rounds.