Brennan had a run of five straight birdies between the fourth and eighth holes, and recovered from a bogey on the 11th by picking up further shots on the 13th and 18th holes, saving par on the 17th with an excellent second shot after running into the rough with his tee-shot.
Hossler, who shared the overnight lead with Brennan, had kept himself in touch with a birdie on the third and an eagle at the par-five fifth, but then bogeyed the ninth and 12th holes as Brennan surged forward.
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"I did great for nine holes," Brennan said. "I felt really good on the front nine. Just playing my game, I was really staying within myself, hitting good shots.
"Felt like I was really in control. Got a little nervous around 10 and 11. I made a really nice par putt on 12 and that kind of steadied me a little bit going through and I'm just glad to make a few putts coming in.
"It's everything. I mean, I don't want to be done playing golf. I feel like I've played the best golf I have all year the last couple weeks.
"Unfortunately, a couple tough Sundays, but outside of that, I've been playing really solidly. This is my sixth week in a row and so seven's a lot, but I think I'm going to be perfectly happy running on fumes next week."
England's Harry Hall finished with a round of 65 to take a share of fifth place, one shot clear of compatriot Matt Wallace who had a share of seventh after he also closed with a 65.
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