But Montgomerie was more pleased with the way he bounced back earlier in the round after a poor start.
"It's never easy leading," he told reporters.
"I was two-over after six (holes) and got back it back to level for the day. It was very easy to let that go today."
Montgomerie secured his first major title on the over-50s Champions Tour at the Senior PGA Championship in May.
Second-placed Dunlap prepared for the Senior Open in an unconventional manner.
He spent last week playing some of Ireland’s great courses, following an unsuccessful quest to qualify for the British Open, and it left him in a relaxed state of mind.
“I have been particularly good with the putter this week, made all my short ones,” Dunlap, 50, told reporters.
He never won on the PGA Tour, but had a solid career, three-times finishing third, including at the 2000 Players Championship.
“At this age, you know, the more I hit balls, the more my left thumb or left wrist I feel ... I have got friends that are the golf equivalent of basketball gym rats. That works for them. It doesn't work for me."
Fiji's Vijay Singh was within striking distance, four strokes behind in just his second start on the Champions Tour.
Singh said he was not quite ready to turn his back on the regular tour, where he has won 34 times.
“I just want to come out here and see how it's like and hopefully take home the trophy on Sunday,” he said.
(Reporting by Andrew Both in Cary, North Carolina,; Editing by Gene Cherry / Ian Ransom)
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