The opening set was a very tight affair, as both players were afforded just one break point each.
Berdych managed to convert his, which was enough to see the 27-year-old Czech go one set in front.
Berdych, who recorded 12 aces, continued to dominate on his service during sets two and three.
Conversely, Melzer began to falter and was broken on four more occasions, which eventually led to his opponent prevailing 6-3 6-2 6-2 in 108 minutes.
Berdych will face Kevin Anderson in round four.