The Briton had taken a four-shot lead into the final round but four birdies from Moore in an inspired 65 put the American in the clubhouse on nine under.
However, Rose held his nerve on his final two holes to clinch the title.
"This was an important day for me," said Rose, who moved back up to 16th in the world rankings following his triumph.
"I knew having not closed out last week it was important for me just for myself to do it today, but to do it in a way I felt like I really did put into play the lessons I learned at Hartford.
"I played much slower and I really felt calm. I didn't feel like the nerves got the better of me at all the whole weekend, so I was much better at that.
"It's still never easy to close these things out. I had it going nicely on the front nine today and probably got a little bit closer than I should have with two slack three-putts 10 and 11, but I knew around level par would get the job done."
Woods had an inconsistent day, carding four bogeys and three birdies, but insists he is playing well ahead of the Open, which starts on 15 July at St Andrews.
I hit driver as many times as I possibly could because it felt so good. I just wanted to keep hitting it. That hasn't been the case lately
"I'm really excited. I've driven the ball better this week than I have in a very long time," said the world number one.
"It's fun to hit the driver that way. I hit driver as many times as I possibly could because it felt so good. I just wanted to keep hitting it. That hasn't been the case lately.
"It was nice to get back dialled in but obviously I need to get my putter working a little bit better and get rolling."
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