Gomez, searching for a maiden PGA Tour victory, shot a seven-under round of 65 at the Trump International Golf Club with his one bogey coming on the 10th hole.
Brown, from Augusta, Georgia, had a five-under 67, also with just one bogey.
Andres Romero, another Argentine who was in strong contention, fell down the leaderboard after a triple bogey on the par-four 12th. He eventually finished at three-over 75, leaving him seven shots off the leaders.
Australian Matt Jones shot the best round of the day with an eight-under 64 that moved him within six strokes of Gomez and Brown.
Gomez said he was hoping for some wind on Sunday.
"There are many good players in this field and it's a course where you can go low. So you have to work hard and play well.
"If it gets windy, that'll be better. It will play a little more difficult. If there's no wind, you will need a lot of birdies to win," he said.