Greg Norman, who is making his Champions Tour debut, fired the round of the day. He carded a 4-under 67 to move into second place at 1-over-par 143.
Stadler, who owns two major championship titles among his eight wins on the Champions Tour, made the only mistake of his round at the par-4 opening hole. He bogeyed one to drop to plus-3.
The 52-year-old got that stroke back with a birdie on the second. Stadler came right back with a birdie on the third on Royal Aberdeen Golf Club to get back to plus-1.
Stadler, the 1982 Masters champion, birdied the par-5 sixth for the second straight round. He calmly ran off eight consecutive pars around the turn.
Stadler then rolled in a birdie putt on the par-4 15th to move to minus-1. He closed with three straight pars to complete his round and was the only player in red figures after the second round.
'Today was very satisfying. I hung in there and I took advantage of my opportunities to make birdies,' Stadler said. 'Four birdies and a bogey around here are pretty good. I just kept it in play and was patient and pretty much got as much as I could out of the round.'
Norman, the 1984 and 1993 British Open champion, got off to a hot start with consecutive birdies from the first. He bogeyed the third for the second straight round.
The Australian got that stroke back with a birdie on the sixth. Norman faltered to another bogey on the par-3 eighth, but quickly got that shot back with a birdie on No. 9.
Norman parred his next five holes. He rolled in a birdie putt on the par-4 15th to move to plus-2. However, he faltered to a bogey on the next. Norman birdied the 17th, then got to plus-1 with a 30-foot birdie putt at the last.
'I got to know the lines on the golf course,' said Norman. 'You know where you're hitting the ball, you know where the trouble is. So I'm starting to feel a little bit more comfortable with it and hopefully I can keep it going the next couple of days. I knew today I could shoot myself out of the tournament and I didn't want to do that.'
Seiji Ebihara shot a 1-under 70 in the second round. He moved to 5-over-par 147 and shares eighth place with Frank Connor.
Mark McNulty, won owns four wins on the Champions Tour, shot a 1-over 72. He stands at plus-6 and was joined there by Norm Jarvis and Horacio Carbonetti.
The cut fell at 15-over-par 157. Senior Players Championship winner Peter Jacobsen and 2003 Senior PGA Champion John Jacobs made the cut on the number.
Among those missing out on the final two rounds were: Tom Kite (plus-18), 2004 champion Pete Oakley (plus-20) and Bruce Summerhays (plus-21).