I don't have a solution... really.
This exact same thing happened to my MS-16F3 (which is the barebones version of the GT60) and it's of course a few months out of warranty. Was playing Watch Dogs, walked awayor a sec, came back, computer's completely shut off. Doesn't respond to power... it's completely dead. Took RAM out and replaced to test, one by one. Took entire laptop apart and noticed the motherboard powers on with nothing attached at all. After playing around with a multimeter and multiple random tests, I believe that it's either the GPU or the motherboard at fault... which sucks because those are the two most expensive parts to replace.
I am running this laptop right now without the GPU in it. The fan is constantly at full blast and it thinks it is running the NVIDIA chip, even without it inserted. I'm beginning to think it's the mobo itself and that my GPU should be fine, because now my laptop won't charge the battery unless the power adapter is plugged in with the OEM charger plug a certain way (there is a black dot on the metal part that has to face upwards), but I can't exactly test without an additional laptop... and I don't have the money to throw it at a mobo right now.
UPDATE: Fixed the stupid charging problem with a piece of foil shoved into the power plug... the center pin wasn't exactly making contact with the laptop.
UPDATE 2: Foil trick did not fix it. There's something wrong with the mobo and I still don't know what it is... may be a short or something causing trouble. I stripped the laptop down to the bare components and tried to turn just the mobo on, no RAM, just CPU. On DC, it doesn't work. With battery, it does. Putting the GPU in next yields nothing at all, even on battery.
DC adapter consistently provides 19.5V, according to my multimeter, so that's still good.
Basically, the motherboard is bad and I will need to get it replaced. As a tech-inclined person who never bought into warranties, I'm now willing to look into getting a crappy SquareTrade one just so I can get my money back if things go south with components. That and I'm not impressed with MSI... but I am with my 6-year Asus G50VT-X5 that's still running.