The San Diegan, with a chance to become the first golfer to shoot 62 in the 157-year history of major championship golf, had an 18-foot birdie putt on the 18th hole at Royal Troon roll around about half the hole before staying out.
The last Open to be held at Troon was in 2004 when the completely unheralded Todd Hamilton came away as the champion.
Montgomerie was 3 under for the front nine and 3 over for the back nine.
Bubba Watson, Justin Thomas and Patrick Reed are tied for the lead at 4 under.
South Africa's Louis Oosthuizen bagged the first hole-in-one of the week when his tee shot at the par-three 14th landed at the front of the green and dropped in. Jordan Spieth meanwhile endured an error-prone afternoon by his own lofty standards, bogeying on the final hole to finish his round even-par 71.
Colin Montgomerie started with a buried lie in a pot bunker on No. 1 and made double bogey.
Heartbreak at Royal Troon for Lefty. Spieth said this again two weeks before the U.S. Open.
With him at the top of the leaderboard were Lee and the American Justin Thomas, who was a model of consistency: he birdied the first, second and third holes. "When I found out I got in here, I'm thinking, 'Hmmmm, that's kind of interesting.' It was exactly what I needed". He faltered in the 10th hole with a bogey while returning back to putt a birdie in the 13th hole.
The laid-back American comes into the championship fresh from finally winning a major at the US Open last month and he shares the status of favorite with world number one Jason Day of Australia. "It's one of these golf courses that it allows you to get off to a quick start and allows you to get nearly overconfident and cocky".
Luke Donald, playing alongside Montgomerie in the first group, was at 2 under.
Asked how he will change his game plan in the wet and windy conditions, the Texan added: "If it's what's forecasted it looks like we've got rain while we're warming up and it might start to diminish as we start and come back in on the back nine with an opposite wind". He might want to heed his own advice.
Prestwick, site of the first 12 British Opens, is adjacent to Royal Troon.
"The score didn't really reflect how well I played", said Rose, whose top-10 finish a year ago was his best showing at The Open since tying for fourth as an amateur at Royal Birkdale in 1998.
Adam Scott, desperate to get his hands on the elusive Claret Jug after four painful near misses in the past four years, tees off just before 2am (NZT). Five dogs were on the beach.
"But I knew today was a day where you had to make the most of the conditions because I don't think we're going to see the course like this for the rest of the week".