I have never seen a power supply turn itself on. That is scary from a safety standpoint. If the new PSU is doing exactly the same thing as your old PSU, I don't think it is your PSUs.
Make sure the case's front panel power button is not stuck pressed in.
Note the time this happens to see if there is a pattern. There are many things that can, for legitimate reasons, wake a computer including Windows Update, security programs, program updaters, and more.
Next time it happens, from an elevated command prompt (run as administer), enter powercfg /lastwake to see if it shows what woke your computer.
Then look in Task Scheduler to see if you can find that program and change its Triggers to not allow it to wake the computer. If you cannot find anything, you may have to go through all the entries to see if anything is scheduled at the time(s) your computer wakes. This can be a long tedious process, especially going through the Microsoft entries.