How to control ESP-05 power using Arduino (both are 3v3 devices)
You're on the right track, but what N-FET did you use?

First: how much power?

The article here suggests you might need to switch 500 mA.  ESP8266EX has a typical transmit current of about 170 mA and no listed maximum.

You don't say how much time you'll have the ESP-05 enabled nor how much power you can allow during its off-time.

Second: What threshold voltage?

Fundamentally an nFET is a voltage controlled switch.  The level at which it switches is the VGS(thresh) and you need one which will switch at 3.3V, often called a "3.3V logic-level MOSFET" (watch out for those which are good for 5V logic).  Many FETs, especially those for larger currents and voltages, have a much higher switching threshold.

Two questions about this specifically:


https://electronics.stackexchange.com/a/106443 
https://electronics.stackexchange.com/a/235680


Sparkfun shows modules for which might immediately do what you need, if you don't mind using a ready-made module.  Seach for MOSFET on that site.


One product there uses FDS6630A which has VGS(thresh) of min=1.0V, typ=1.7V, max=3.0V.


Once you've chosen a MOSFET, you look for examples for the details of its required current limit and bias resistors.  (You won't need the diode of your example circuit, that's for the back-EMF of the inductive load.)  The FDS6630A example above uses


Source: Sparkfun documentation

Other possibilities

SLEEP The ESP8266EX has various sleep modes its various sleep modes is 15 mA down to 0.5 &mu;A according to its datasheet.  You can enable these with AT commands, and perhaps that will work for your application.

RESET Some ESP-05 models have a 5-pin connector with RESET signal.  If you assert this, I'll guess the power consumption will be very low (but couldn't find a value in the documentation) and perhaps that would work for you.




Article about ESP-05 https://blog.startingelectronics.com/esp8266-esp-05/
ESP-05 chip is ESP8266EX: datasheet
ESP8266 AT Commands manual