Both players picked up a break in what was a tight opening set, but Berdych stepped up a gear when it mattered, storming through the first set tie-breaker.
The Czech maintained a higher level in the second set, producing significantly more winners and claiming 87% of points on his first serve.
However, Melzer had his chances, failing to convert any of the three break points that came his way, while Berdych converted two out of three.
Melzer made more errors in the third set as he looked to force his way back into the clash, allowing Berdych to claim the final set without doing anything too special.
Twenty-fourth seed Richard Gasquet of France set up a third round tie with Kevin Anderson of South Africa after defeating James Duckworth of Australia 6-2, 6-3, 7-5.