The Uruguayan had lost his last previous two encounters with Melzer, but he levelled their head-to-head record at 2-2 at the ATP 250 hard-court event in Kuala Lumpur.
Cuevas started the 2014 season outside of the world's top 200 players, but titles in Umag and Bastad earlier this year have helped him climb to world number 37.
The 28-year-old showcased his defensive qualities against Melzer, fending off four of the five break-point chances the Austrian orchestrated.
The fifth seed also fired down eight aces on his way to victory in a time of one hour and 41 minutes.
Cuevas now meets Ivan Dodig in round two after the Croat defeated Hiroki Moriya 7-6 (8-6) 6-1.
Rounding off Monday's first-round matches, Serbia's Filip Krajinovic saw off the challenge of fellow wildcard Taro Daniel 6-2 6-4.
The likes of US Open runner-up Kei Nishikori, world number 13 Ernests Gulbis and top-25 player Leonardo Mayer will be in action later this week.