England's Lee Westwood (72) is in a group on four under alongside America's Hunter Mahan (68), Ireland's Padraig Harrington (68) and Sweden's Henrik Stenson (70).
Tiger Woods will begin the final round 12 shots back after treading water at three over with a level-par 72.
Rory McIlroy slumped to a 77 to end one over and world number one Luke Donald took 75 to end seven over.
Halfway leaders Fred Couples and Jason Dufner fell away on Saturday with rounds of 75 to end two under in a goup which includes England's Ian Poulter, who carded 70. Scot Paul Lawrie has sole possession of 10th place on three under after a 72.
Eight players held at least a share of the lead at some stage on an absorbing day in the Augusta sun.
Despite favourable scoring conditions, no-one was able to escape from the pack until Mickelson sank a 25ft eagle putt on the 13th to join Kuchar and Oosthuizen in the lead at six under.
That seemed to inspire Hanson, who was playing up the 14th, and the 2010 European Ryder Cup player made four birdies in his last four holes.
Mickelson, who made his first birdie at the 10th and added another at the short 12th, was energised, too.
He fired further birdies at the par-five 15th, courtesy of an outrageous, high-risk high pitch from off the green to a few feet, and a birdie for a third day in a row at 18 after hooking his second shot around trees onto the green.
Mickelson's majestic back nine
Mickelson credited Thursday's fightback from four over after 10 holes to end with a two-over 74 as the foundation for his bid to equal Woods and Arnold Palmer's record at Augusta.
"When I look back at this tournament I'll look at Thursday," said the Californian, who has won all of his previous four major titles from the last pair.
"At some point I'm going to get hot here but I've got to be in a position to move up the leaderboard, not just get into the weekend. The way I fought back Thursday made that possible."
"It's going to be tough. It's a new situation to me," said the Swede, who lost 4&2 to Mickelson in the 2010 Ryder Cup singles at Celtic Manor.
"I'll try to do the normal stuff. I've got my two kids here and my wife, so just try to stay in the house and cook in, and like I said, I probably won't be watching a lot of Golf Channel or stuff. Just trying to stay away from all that and try to get as much sleep as I can and try to be ready."
Oosthuizen, who has struggled with injury since his Open triumph at St Andrews, joined Mickelson on eight under on the 17th but a wayward approach into the crowd handed it back on 18.
Wayne GradyGolf Analyst, BBC Sport
Hanson played magnificently - every tee shot, he was picking up the tee straight away and his iron play was spectacular. He was very much in control but there's a difference between having a low round and taking the lead into the final round. He'll have a sleepless night tonight. He will be playing with Phil Mickelson, a three-time champion, but the expectation Phil has on himself is pressure itself, while Oosthuizen is just sitting back there.
He fired birdies at the par-four third and the short fourth but soon dropped a shot at the par-three sixth courtesy of a three putt and again at the par-five eighth and was unable to take advantage of any of the holes coming in.
"It's so frustrating because I am so close to putting it together," said the former world number one. "I unfortunately did not play the par fives very well. I just didn't take care of the opportunities when I had them.
"I would just hit a couple of good shots in a row and compound the problem with a bad shot in the wrong spot."
The 22-year-old and playing partner Sergio Garcia, who shot 75 to end one under, acknowledged their shared struggles with a hug on the 12th green after both making their first birdies of the round.
"It was a congratulatory hug," said McIlroy. "We couldn't feed off each other's good energy because there wasn't any," added Garcia. "Our bad holes were really bad and our good holes were bad."
Westwood, who was second to Mickelson in 2010, birdied the second to edge to five under for a share of the lead but dropped back to two under after 13 before clawing his wasy back into a tie for sixth with birdies at 14 and 15.
Hanson and Mickelson will go out in the final group at 1440 (1940 BST) on Sunday as the 76th Masters nears its conclusion.
Harrington ends round on high
THE MASTERS ON THE BBC
Live TV coverage: BBC Two from 1900 BST; Amen Corner plus holes 15 & 16 from 1630 BST on Red Button & online
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