Right click on computer, settings, advanced system settings, startup and recovery, settings, turn off automatic restart if it is checked.
Also, even if the heat numbers seem low, it would be nice if you could take the side of the case off and point some sort of serious fan at it if you have one, like a regular AC oscillating fan.
I updated the OP. Completely forgot to add the PSU in! It's a Corsair Enthusiast TX750. And what would using one of those fans show me? Would I then test the games again to see if the shutdown happens?
Also, I recently unchecked the automatic restart but haven't played a game since then. Would that stop the power off?
try msi afterburner for fan control. I had very same issue, although my gpu temp was through the roof (doesn't seem like the case with you), after i started using afterburner it kept correct enough temp to keep playing without problem. Also maybe check in bios if you have some kind of automated shut of if cpu reaches certain temp.
I finally solved my problem, and it ended up being an extremely easy fix that made me feel stupid.
I had 6 components plugged into one rather cheap power strip, and it turned out that my PC wasn't getting enough power to support gameplay. Since I separated things to different power outlets on the wall, my problem hasn't occurred again.