Aha Seasonic SSP GT - that's why it runs, I've seen tests which showed it actually often run better than 350W and Seasonic are typically excellent PSUs.
Whilst I'm much more assured knowing it is Seasonic, I would also make the note that it is a discontinued product of theres, and I would suspect out of warranty (5 years?).
What I would add however, is I believe the SSP 350W does not have any direct PCIe connections, so I would suspect your seller is using adapters, which is never a good sign.
Using adapters (especially on GPUs) is usually a sign that the PSU wasn't made for that purpose.