MAX7219 powered on via digital pin set to HIGH is not working good (I < 25mA)
After some research I eventually solved my problem, so here's the answer.


  
  Why does it happen?
  


This LED display draws much more current than I measured on Vcc line between Arduino and display because of ghost powering via signal lines (DIN, LOAD, CLK). The display (although slightly dimmed) can even run without Vcc line connected! Real current draw was about 30-40 mA with peaks up to 100 mA.


  
  Is there a way to solve the issue without using a transistor / MOSFET / relay (I suppose it should obviously help).
  


Yes, by limiting display current draw.

For an unknown reason these cheap MAX7219-driven LED boards has very small current control resistor RSet of 10 kOhm, resulting in huge Iseg ~= 40 mA. Replacing it by something like 62k will limit per-segment current to ~10 mA.

This is what the datasheet says:


And this is how it can be done:


Works perfectly! Now I can power off LED display completely, so it consumes no current when the device is sleeping.

NOTE: it's important to properly re-initialize MAX7219 after power on!

In the colorized table above one could find display current draw for different software-set intensity settings in case of using RSet = 62k. GND line values (blue column) seems to be correct as they well correspond with overall device's current draw (the most correct measurement, I suppose).