He was tied 68th.
Martin Kaymer of Germany led the field with a 65. He had a great stretch over back nine with four birdies.
In all, he had six birdies and just one bogey on seventh. He was three clear of Kevin Na, Graeme McDowell, Brendon de Jonge and Fran Quinn, all of whom shot two-under 68. Phil Mickelson carded even par 70, while Adam Scott (73) and Bubba Watson (76) were way down.
Asian Tour star Thongchai Jaidee of Thailand also opened with a 73 as did another EurAsia team member Kim Hyung-sung of Korea.
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Kapur, making his debut at the US Open after coming through the qualifiers, was steady for the first eight holes, with one birdie on the 13th and no bogeys.
Then he bogeyed twice in succession on 18th and first despite fine drives. The second shot, which is so crucial at Pinehurst was where he faltered. On 18th he went left and missed par and on first his second shot went into greenside bunker and he was unable to get the up and down in.
"I was quite fine for the first nine holes, but missed pars on 18th and first despite getting off well from the tee. The second shot is where I went wrong. The bogey on fifth and ninth were disappointing. Overall at 73 I have a lot of work to do in the second round," said Kapur, who has played at the British Open twice before.
Noh Seung-yul of Korea, who holds the distinction of being the youngest Asian Tour Order of Merit winner, enjoyed a good start in the year's second Major when he shot a 70 to take a share of 16th place, while Liang Wen-chong, the first Chinese to win the Asian Tour Order of Merit in 2007 posted 74s and Y E Yang carded a disappointing 75.