Brown birdied the final hole to Gomez's bogey, the swing presenting the title to Brown after a mid round slump had threatened to bring his chances undone.
After a run of three bogeys in four holes through the middle of his round Brown bounced back with birdies at the 12th and the 18th to take his first PGA Tour title.
Brown played the PGA Tour for the first time last season but lost his status and was unable to regain it via the Tour School. Now his worries are over, the victory providing a two year exemption.
Argentine, Gomez, returned to the PGA Tour in 2013 after finishing 10th at the Tour School last year and although the narrow loss is agonising he will earn a big cheque to go towards his quest for ongoing status.
Speith has set the mark for Brown and Gomez to beat when he birdied his final hole in the group ahead but he would have to settle for a share of second position.
Speith was one of America's best amateurs before turning professional. He was a multiple winner of Junior USGA Championships, a Walker Cup team member and the world number one amateur before turning professional last December.
Cameron Percy and Matt Jones tied as the leading Australians in 14th position while Nick O'Hern had a good week when he finished 18th.