McDowell, returning to tournament golf and his European Tour roots for the first time since his win at the U.S. Open, shot a 71 on a blustery day at Loch Lomond.
He openly blamed some of the victory parties he has attended in the past two weeks.
"There were a few good swings in there and a few champagne swings as well."
McDowell birdied his final two holes to finish alongside Phil Mickelson, who was making his first appearance in Europe since winning the Masters in April.
They were both 44 places down the leaderboard behind Clarke, the former Ryder Cup star who has considered retirement after slipping to No. 179 in the rankings.
"I have been frustrated," admitted the 41-year-old Clarke. "I have been telling everyone I have been hitting the ball well, but I haven't been able to hole putts.
"But at the French Open last week I changed my setup with my caddie Brendan McCurtain and started rolling the ball better again."
Clarke had six birdies and no bogeys despite playing in the afternoon, when Loch Lomond was hit with heavy rain showers. He missed two birdie chances inside 10 feet on Nos. 15 and 16.
One shot behind Clarke were England's Graeme Storm and Italy's Edoardo Molinari, who both shot 66 in milder morning weather, and Irishman Damien McGrane.
With the British Open at St. Andrews next week, the Scottish Open attracted several other American players besides Mickelson. John Daly, who won the British Open in 1995, was just two shots off the lead after a 67.
In the group of six players alongside Daly at 4 under was Japanese teenager Ryo Ishikawa.