Lyle and Park finished the two-day tournament atop the leaderboard at 6-under 138 after rounds of 71 and 70 respectively at the Sentosa Golf Club's Serapong course.
'It's a dream come true,' Lyle said. 'I can't wait to get over there and see what's it's all about.'
Kapur needed four sudden-death playoff holes to beat Lee Sung-man of South Korea and secure his place at the world's oldest major, which will be played July 20-23.
Park had three birdies -- including a eight-footer on the 18th hole.
'I almost closed my eyes when I hit my last putt,' Park said. 'It's a milestone for me. I've always wanted to play in the big events.'
Kapur, last year's Asian Tour Rookie of the Year, shot a 70 to finish the qualifier tied with Lee (68) for the third and final spot at 139.
'It's pretty cool to be in the Open, it's been great this past one year,' said Kapur, who made a birdie on the 18th hole to force the playoff. 'It was a bit of hard work but I think preparation like this will make me stronger.'