Johnson was 5-under through a five-hole span of his first seven holes and had a birdie, one bogey and nine pars the rest of the way for his tournament-leading round.
Johnson is one stroke ahead of Rafael Cabrera-Bello and Mark O'Meara, who shot 4-under 67. O'Meara's round including five front-side birdies and eagle at 17 and three bogeys. Cabrera-Bello had three birdies and a bogey on each nine Thursday.
The six-player group at 2-under included Phil Mickelson, who had a bogey at 18, and 19-year-old Jordan Spieth, whose win Sunday at the PGA's John Deere Classic earned him an invitation to the British Open.
World No. 1 Tiger Woods, who had a very poor tee shot at the first hole, put together three birdies over Nos. 10-13 and also wound up at 2-under.
Conditions at the Muirfield Golf Links in southeastern Scotland were very good, with relatively light winds and temperatures in the upper 70s. But those conditions caused problems for golfers with later tee times, as putting became more difficult as the day wore on.
Kapur was one of the few late starters -- Woods was another -- to be among the 20 golfers who shot better than par and he did his good scoring early in his round. He opened with three birdies and strung together three more over Nos. 5-7 to take the lead. But his advantage lasted just three holes as Kapur had a double-bogey at the 10th and a bogey at No. 14.
He went out in 30 and then had a 38 on the second nine.
Zach Johnson posted his first birdie at the third hole and really became a factor with a long eagle putt at the fifth. He followed that with birdies at Nos. 6 and 7 to reach 5-under. He was at 6-under after a birdie at 12 but dropped back with a bogey at 14 and had pars the rest of the way in.
No. 2-ranked Rory McIlroy battled through an 8-over 79. His round included birdies on two of the pars 3s but also two double bogeys and six bogeys.