Garcia came up three shots shy of extending his run at the major into a 26th successive year.
Garcia was unable to claim one of the five places on offer in the 36-hole qualifying event at West Lancashire.
The New Zealander claimed his first DP World Tour title after recording two eagles and a birdie in his final four holes.
England’s Aaron Rai is tied for fourth on 17 under.
Andy Sullivan, Oliver Wilson and James Morrison are in a six-way tie for the lead on seven under.
They lead by one stroke from America’s Rickie Fowler and Sweden’s Ludvig Aberg.
Rose’s 73 left him joint top of the leaderboard alongside Antoine Rozner on six under.
The hole of the day came from American Dylan Wu who landed an albatross on the 14th hole, needing just two strokes on a par five.
Rose carded eight birdies and a solitary bogey to record a seven-under-par 65.
The top-ranked player in the field carded eight birdies and a solitary bogey to record a seven-under-par 65 at The Belfry.
