The percussionist was performing with his band OHM at The Baked Potato jazz club in Studio City, California, where he suffered a "massive heart attack" just three songs into the set, his manager Rob Bolger told the BBC.
Paramedics were called and he was admitted to a nearby hospital, where he was subsequently pronounced dead.
The tragic news has hit Mustaine hard, and he took to Twitter.com on Sunday (22May16) to express his disbelief.
"TELL ME THIS ISN'T TRUE!," he exclaimed. "I woke at 4 AM to hear Nick Menza passed away on 5/21 playing his drums w/ (with) Ohm at the Baked Potato. #nickmenzarip".
He later added, "I am gutted".
Menza, 51, served as the third drummer for MEGADETH from 1989 to 1998 and performed on their albums Rust In Peace, Youthanasia, Cryptic Writings, and their classic Countdown To Extinction.
He also played on then-Megadeth guitarist Marty Friedman's solo projects, Scenes, Introduction and True Obsessions, before leaving the band in 1998 after he was diagnosed with a tumour on his knee.
Friedman, who exited Megadeth in 2000, has also mourned his pal's passing on social media.
"We all know the great and unique drummer that Nick Menza was, but he was also a trustworthy friend, a hilarious bandmate, as well as a very loving dad," Friedman wrote on Facebook. "I'm beyond sad, did not see this coming at all. RIP Brother."
One of his OHM bandmates also paid tribute to Menza online, writing, "RIP Nick Menza. In the brief time I knew you, it was a pleasure. We will miss you 'MAChine'".