Gomez had a one-shot lead coming into the 18th on the Composite Course at Trump International Golf Club, but stumbled down the stretch, ending with a final-round 71.
Meanwhile, Brown -- playing on conditional PGA Tour status -- came through with a clutch birdie to card a 2-under 70 and finish 72 holes at 20-under 268.
Brown's initial victory enabled him to regain his tour card and land a spot in this year's PGA Championship.
Gomez, who came within one hole of his own first career win, settled for a second-place tie with 19-year-old Jordan Spieth (67) at 19-under.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys