I likewise could not find any information re. that PSU.
Try loading HWInfo64 and run that program; take a look at the +12volt, 3.3v, and 5volt readings; the program can also log to a file so you can start the program and select logging then start playing a game, to warm up the PSU. Once you hear the PSU getting noisy, before the PC crashes, close the game and let things cool down and review the log.
The PSU should be able to keep the power within 10% or better; if it can't or doesn't then it is definitely time to replace the unit. Please post the results.
Also, you didn't post the specs from the label for those voltage rails; If that power supply is an older design it may not have enough power (amps) on the +12volt rail to run your system under a heavy load (gaming). And, of course, there is the real probablility that it is just worn out and something inside is failing or has failed so it is struggling just to produce any power.
If you decide to simply replace the PSU with a new unit I suggest you select a model from tier 1 or 2 of this list:
http://www.tomshardware.com/forum/id-2547993/psu-tier-list.html
They might be expensive, but the system you save might be your own.
The minimum requirement for a
reference R7 370 is a 450Watt psu with 33 amps on the 12volt rail: this is for your entire system. A reference or stock 370 supposedly has a max power draw of 110w or about 9 amp. The MSI is not a stock unit but has been upgraded with better cooling and from Newegg's description may draw 150watts max or about 12.5 amps. I always like a little headroom with my power supplies so I would recommend a 500-600watt unit - in part because there was a test run on Tom's (can't find it right now) that showed video card power spikes, while in gaming, quite a bit higher then the advertised/published maximums.
Hope all this helps.
EDIT: I was writing this when DonkeyOatie found the information re. the PSU. If it is a CiT unit, replace really soon before something terrible happens. Don't even worry about the monitoring info I talked about.