Gil Morgan carded an even-par 72 and is alone in third place at 6-under-par 138.
Hard, swirling winds greeted the players Saturday morning and scores reflected the difficulty. Only nine of the 77 players in the field broke par and only three of those rounds were sub-70.
On Friday, Koch established a new course record with a 10-under 62. Conditions were benign and there was no wind but things changed for the players right away on Saturday.
'It was quite a bit different than yesterday,' said Stadler. 'I hit a lot of good shots today. The long irons were pretty sporty in the wind. I'm usually okay in a breeze but I'm not really that good in winds like this.'
Stadler, a three-time winner on the Champions Tour last year who became the first player to come back to win on the PGA Tour when he titled at the B.C. Open, began the round five strokes behind Koch.
Stadler knocked a 5-iron to three feet to set up birdie at the second. He added an eight-footer for birdie at the par-5 sixth to inch his way closer to Koch's lead.
He missed the green with a 4-iron at the par-5 10th but chipped in from 30 feet to get within one of Koch. Stadler then hit a nine-iron to eight feet at the 15th to collect his fourth birdie of the round and match Koch in first place at minus-nine.
Stadler reached the green in two with a two-iron at the par-5 17th and two- putted from 40 feet for birdie and the lead.
'It was a pretty good day of survival,' admitted Stadler, whose son Kevin held the first-round lead at the PGA Tour's Buick Invitational this week. 'I took advantage of some holes downwind and made some birdies. I had a few good breaks and I had lots of chances.'
Koch is still in the hunt for his first victory since joining the elder circuit last year. His round started with a bogey at the first when his second shot missed the green and he failed to convert a 10-footer for par.
The NBC commentator held his own against the demanding conditions with 12 consecutive pars, including a great save at the 11th when he blasted out of a bunker to six feet and rolled home the putt.
Koch made another mistake at the 14th when he nailed a 5-iron through the green. He had a long way to go between his ball and the hole and his chip came up 12 feet short. Koch failed to save his par and fell two behind Stadler, who was already in the clubhouse.
Koch got one back with a long birdie putt at 16 and parred out to polish off his 1-over round of 73.
'I am satisfied with 73 and where I am after two rounds,' said Koch, who is looking for his first win on the 50-and-over tour. 'It's been a long time since I've been in this position.'
Tom Watson carded the second lowest round of the day Saturday with a 3-under 69. He moved into a tie for fourth place Wayne Levi (71) and Morris Hatalsky (73) at 5-under-par 139.
Lonnie Nielsen, who was in second place at the start of the second round, struggled to a 5-over 77 and shares seventh with D.A. Weibring (73) and Larry Nelson (75). That group is at 4-under-par 140.
Jack Nicklaus, the course architect of TwinEagles, shot a 2-over 74 and is part of a logjam tied for 15th at minus-2.