"I was just lost for words," he told Newstalk ZB.
"I managed to play really well in the first round, held on for the front nine in the second round and then made a few birdies coming home. Finished nicely with a long putt on the last. I'm really excited to get it done and looking forward to getting to Royal St George's in a couple of weeks," Hillier said.
Hillier has been playing on the second-tier European Challenge Tour with two top-10 finishes this season.
The other two New Zealand players have had to wait more than a year after earning their qualification places after the 2020 British Open was cancelled.
Fox booked his place after leading the order of merit on the PGA Tour of Australia following the 2019 season.
Lee meanwhile qualified last March after a top-10 finish at the Arnold Palmer Invitational.
Fox will go into the event in decent form, having claimed three consecutive top-15 finishes on the European Tour, but Lee has been struggling, with a best finish of tied 19th in his PGA Tour season.
Lee sits 171st on the FedEx Cup standings, and needs a top finish in the seven remaining events to move into the top 125 and secure his playing card for next season.