Kapur, who was three-under for the day through 15 holes, made a triple on the par-4 16th, but kept control of his nerves to finish with pars and a card of even par 70 after a first round 73.
At three-over 143, Kapur was tied 33rd to become only the second Indian, after Jeev Milkha Singh to make the cut in an US Open.
Kapur tied 68th after the first round, where he bogeyed three times on the back to finish with 73, rose to 35 places and was among the big movers on second day.
Then on the long 532-yard par-4 16th, he found the fairway bunker from the tee and was unsuccessful in getting out of it with a 5-Wood. He chipped out but only to get into a greenside bunker and then missed an up and down for a seven.
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Kapur's birdies came on the third where he took advantage of the tee being moved up and he got close to the green with his tee shot. His lob wedge landed within two feet for tap-in birdie.
Then came the triple bogey, but he handled it well and did not let it disturb the day.
The cut fell at five-over-145. Sixty-six professionals and one amateur made the cut.
It was the second cut in a Major for Kapur and he is in a position to improve on the best finish by an Indian, the 36th place finish by Jeev in 2007.