The 22-year-old, who carded a two-over 73 in the opening round, took his two-day total to two-over par 144 as he is to make the halfway cut which is projected to be at three-over at the Carnoustie Golf Links.
Shubhankar struggled in the front nine, scoring three bogeys on the first, eighth and ninth holes but made a strong comeback in the back nine.
He got a birdie on the 10th hole before getting a bogey on the 12th hole, but came up with two consecutive birdies on the 13th and 14th.
He completed the comeback with a birdie on the 18th and final round to earn a 71 on the second day.
"I was trying to be as comfortable as possible out there and try to be as free as I can. But deep inside, I was very nervous. I struggled with every shot. At the end of the day, you got to tell yourself that you're playing your own game and not think about playing in a Major championship. If you think about that, you will mess it up," he said in an Asian Tour release.
"This is my favourite Major championship. I am happy to have made the cut now. This is also the first time I made the cut in a Major. This is definitely a big moment for me and it's also my 22nd birthday tomorrow. It will be a great birthday present for me for sure," he added.
"It has been great this week. I also imagined myself playing in The Open since I was a kid. This was one thing that I really want to do, to tee up and play in front of these crowds. I am really happy."
Americans Zach Johnson (four-under 67) and Kevin Kisner were at the top with a total of 136, while legendary Tiger Woods got an identical 71 for the second round in a row to make the weekend cut.
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