TROON, 16 July 2016, (AFP) - Bidding to win his first major at the age of 40, Sweden's Stenson played some superb golf for the joint-lowest round of the day at Royal Troon and sits at 12 under par overall.
Mickelson, who had been in front after each of the opening two rounds, didn't do too much wrong as he shot a one-under-par 70 to sit at 11-under altogether.
The pair are are now poised for a final round shoot-out for the Claret Jug with American Bill Haas third but well off the pace at six under par.
Stenson followed his 65 on Friday with three birdies in the opening four holes on Saturday, as the wind blew strongly on Scotland's west coast but the rain mostly stayed away.
He dropped back with bogeys at the long sixth and again at the short eighth, where he found himself in a bunker off the tee.
But he played some brilliant golf on the extremely challenging back nine, going two-under coming home with birdies at the short 14th and 17th holes.
"It was tough out there again, even though as normal the front nine gave you some more opportunities and I got off to a flying start, hit some great shots, took some opportunities," said Stenson.
"I easily could have been four-under through five and then dropped a couple of shots.
"The key moment was really 10. I made a 35-footer for par to stay in the game. It felt like Phil had a bit more momentum at that stage, and then we both had to scramble on 11 and 12, and then I just played really solid from there on, the last six or seven holes."
Mickelson had led by a shot from Stenson overnight and the 46-year-old left-hander played well for much of his third round, with birdies at three, 13 and 16.
He survived awful tee shots at the extremely tough 11th and 12th holes to rescue a pair of par-fours, but bogeyed the same 14th and 17th holes where Stenson flourished.
As a result, the American handed his rival the initiative and gave himself work to do as he attempts to become the oldest winner of the Claret Jug since Tom Morris Senior down the road at Prestwick back in 1867.
"I was off today. I didn't have my best stuff. My rhythm was a little quick from the top as we started downwind. I was a little bit jumpy and my rhythm wasn't very good today," Mickelson said.
"I found a way to kind of settle in and hit some shots and then find ways to make pars on the times that I hit some poor shots.
"Today could have been a day that got away from me, instead I shot under par and kept myself right in heading into tomorrow's final round, so I'm proud of that."
While there has never been a Scandinavian winner of a major, there has been an American champion in each of the last six Opens played at Troon.
That is a good omen for Mickelson, who beat Stenson to win the 2013 Open at Muirfield, but it will also provide encouragement to Haas.
The son of former US Ryder Cup star Jay Haas has never managed a top-10 finish at a major but he has gone under par every day at Troon so far and maintains an outside chance six shots off the pace.
Haas had a 69 on Saturday, the highlight of which came at the Postage Stamp. The short eighth was playing at just 100 yards in the third round and Haas chipped in from a greenside bunker for birdie, one of five on the day.
England's Andrew Johnston is at five-under, while JB Holmes of the United States sits at four-under after a 69.
Elsewhere, reigning champion Zach Johnson saw his challenge fade after a four-over-par 75 dropped him back to one-under altogether, and Rory McIlroy's prospects also ended with another difficult afternoon.
The 2014 champion had a two-over 73 to fall to even-par and his frustration was summed up when he destroyed his 3-wood after a poor approach shot at the 16th.
None of the 'Big Four' have been able to make their presence felt this week -- US Open champion Dustin Johnson is one-under, world number one Jason Day sits at one-over and Jordan Spieth is five over.
Collated third round scores in the British Open at Royal Troon Golf Club, Scotland (GBR&IRL unless stated, par 71):
201 - Henrik Stenson (SWE) 68-65-68
202 - Phil Mickelson (USA) 63-69-70
207 - Bill Haas (USA) 68-70-69
208 - Andrew Johnston 69-69-70
209 - JB Holmes (USA) 70-70-69
210 - Steve Stricker (USA) 67-75-68, Tony Finau (USA) 67-71-72, Soren Kjeldsen (DEN) 67-68-75
211 - Patrick Reed (USA) 66-74-71, Sergio Garcia (ESP) 68-70-73, Charl Schwartzel (RSA) 72-66-73, Keegan Bradley (USA) 67-68-76
212 - Tyrrell Hatton 70-71-71, Jim Herman (USA) 70-70-72, Dustin Johnson (USA) 71-69-72, Kevin Na (USA) 70-69-73, Zach Johnson (USA) 67-70-75