With most of the top players taking the week off ahead of the US Open, 79th ranked Andreev grabbed the spotlight by brushing aside the 17th ranked Melzer to notch his first win since Wimbledon.
The loss added to a miserable North American hard court campaign for Melzer, who also fell at the first hurdle in Washington and last week in Cincinnati.
The Cypriot struggled to find the box with his serve, but was nerveless around the court as he saved all but one of the five break points earned by Sweeting during the match.
Sweeting's second serve let him down repeatedly as he went out of the tournament.
Nishikori, who came through qualifying, was served out of the match by Muller in the opening set, but dug in impressively over the next two sets to earn a second-round clash against Spain's Pablo Andujar.
Dudi Sela , Julien Benneteau, Steve Darcis and Carlos Berlocq also came through opening round matches, with Berlocq the beneficiary when his opponent Andreas Haider-Maurer was forced to retire while trailing 7-5 4-1.