The American is now sixth in the world rankings.
The world number 89 blamed a new putter as he laboured to a closing round of 77 at the British Masters.
Rahm replaced Rory McIlroy at the top of the rankings following his victory in the Memorial Tournament.
Seven years of hard work paid off in spectacular fashion for Michael Thompson at the 3M Open on Sunday.
Thompson carded birdies at the 16th and the last to secure his second PGA Tour win of his career.
Tom Lewis, the only Briton to make the cut, will start the final round eight shots off the lead.
The 23-year-old won a second European Tour title.
It was a second European Tour title for the 23-year-old.
The 29-year-old chose to miss the first six events of the PGA Tour to stay with his family in England.