Bhullar played his first round on the East Course and will be hoping to make up when he tees off at West course in the second round.
Bhullar, the lone Indian in the field, had a birdie on second but gave it back. A second birdie on sixth kept him under before he dropped two shots on 10th and 11th. He fought back on the last five holes with two birdies to ensure an under-par round of 71.
Three players, who have not already qualified and who will finish in the top 10 at the Joburg Open, will secure places in The Open Championship at Royal Liverpool from 17-20 July.
In the event of a tie, the qualifying place(s) will go to the player with the highest ranking in the Official World Golf Ranking at the start of that week.
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Edoardo Molinari of Italy, Alastair Forsyth of Scotland and South African Justin Walters shared the lead with cards of 64 each at East Course.
The field are split between the East and West Courses for the first two rounds, before the action reverts to the East lay-out for the weekend.
Italy's former Ryder Cup star Edoardo Molinari continued to put his recent injury troubles behind him by moving into lead at seven-under par.
Most of the first round leaders played at the traditionally easier West set-up.