In 1988, as a 17-year-old amateur golfer, Rose managed a fourth-placed finish at The Open at the Royal Birkdale.
He finished runner-up in 2018 and will compete at Portrush next month.
After missing out on final qualifying by just one shot in the previous two years, Wheeler was able to get across the line as he posted the second-best round of the day.
“I was over the moon and with it being at North Hants I had a decent home advantage,” he said.
“After the front nine I was three under and I went into the back nine quite comfortable and I played my own game, I made a three-footer on the 18th and the tension just released – I was buzzing.”
Final qualifying will take place on 1 July at four courses where players have a chance to book their spot at the 153rd Open.
Thirty-six holes at Burnham & Berrow, Dundonald Links, Royal Cinque Ports and West Lancashire will decide the final places.
“I want to turn pro eventually and play golf for a living, that’s my long-term ambition,” said Wheeler.
“Hopefully making the final qualifying is one step closer to making that dream a reality.”
