The Northern Irishman led the tournament from day one to wrap up the third major win of his career, following victories at the 2011 US Open and 2012 US PGA.
"It feels incredible. It wasn't easy, there were a few guys making runs at me and I just needed to stay focused. To win three legs of the Grand Slam at 25 is a pretty big achievement," BBC Sport quotes the 25-year-old as saying.
"The lead never got less than two. I always felt I had that little bit of a cushion. I knew I had some holes where I could make birdie and 16 was the real hole for me which I think settled the championship.
"This is the first major my mum has been to that I've won, so mum, this one's for you. I just can't wait to get back and defend this thing at St Andrews next year."