Women's top seed Lindsay Davenport also advanced, wearing down compatriot Meghann Shaughnessy 6-2, 2-6, 6-2.
Sixth seed Henman and Melzer found it difficult to maintain any rhythm in the blustery conditions, with wind gusts reaching 40 mph at times during the match.
With the players trading eight service breaks, Briton Henman admitted he had to battle the weather as well as his opponent on Monday.
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"It was a question as to who would get frustrated first. In those types of conditions it's very difficult to play without mistakes or without your opponent having opportunities.
"You have to accept that it will be ugly at times and find a way through it."
Henman will play 12th seed Tommy Robredo of Spain, a 4-6, 6-2, 7-6 winner over Sweden's Thomas Enqvist.
Davenport and Shaughnessy also contested a scrappy affair as the two hit 51 unforced errors between them.
In the end, Davenport negotiated the tricky conditions better to come out on top.
In other matches, Frenchwoman Mary Pierce ended the run of 15-year-old Czech Nicole Vaidisova 6-4, 6-4.
Third seed Andy Roddick, fifth seed Guillermo Coria, ninth seed Andre Agassi and women's third seed Maria Sharapova will play later on Monday.