Norman, making his debut on the over-50s circuit, came surging up through the field in the evening with a four-under-par 67 - the best score of the wind-blown tournament so far.
Playing amazingly well for a man who had major back surgery a little over four months ago, Norman moved to one over par at halfway and stopped Stadler from threatening to make it a one-horse race over the weekend.
“The difference today was that we just got used to the wind. It wasn’t as gusty and strong as it was yesterday but it was still up there around 20-25 mph but I got to know the golf course, got to know the lines were you’ve got to hit it,” said the Australian.
“I hit a lot more good shots too, which is helpful. Controlling the distance is crucial around here and just staying patient is also very important.”
Norman shot seven birdies in all, holing from 15 feet for a two at the short 17th and 30ft across the last green, having also carded three bogeys.
A few hours earlier, the 52-year-old Stadler, affectionately known as the Walrus, became the first man to break the par of 71 when he shot a 68 to go clear of the field at one under.
The American had four birdies in his round, spoiled only by a bogey at the first, and afterwards said: “I hit the ball much the same as I did in the first round. The difference for me was that I didn’t make the same mistakes as I did on Thursday.
“I’m not playing great but I’m not playing badly either. I’m learning every time I go out to play on this golf course. You’ve got to keep your wits about you. You’ve got to be patient.”
Playing partner Des Smyth, also a successful competitor on the US Champions Tour, although he still lives in Ireland, is third after a 72 left him three over with Tom Watson a further shot back after a 71.
“Every day just proves that Balgownie is a very difficult golf course if there is any kind of wind blowing,” said the American, winner of this title in 2003 at Turnberry after a play-off against Carl Mason.
“Craig Stadler’s 68 was a terrific round of golf under these conditions and if I can get the putter working, there might be a 68 in Tom Watson over the weekend too.”
A stroke further back are Watson, Argentina’s Eduardo Romero, overnight leader Loren Roberts of the USA and Italian Giuseppe Cali. Scotland’s leading player is John Chillas, who birdied the last to finish with a 74 and a share of 13th place with England’s Mark James, American Alan Tapie and South African Bobby Lincoln, all on 149.
A total of 79 players 15 over par or better made the cut but those who missed out included pre-tournament favourite, Tom Kite, after an 83 and 77, and defending champion Pete Oakley, who finished 20 over after a second round 80.