Attiny85 route vcc though output pin
What you want is a P-channel MOSFET to control the power. Something with a nice low "logic level" gate threshold.







simulate this circuit &ndash; Schematic created using CircuitLab

R1 keeps it switched off.  Set the output to LOW to pull the gate down and enable the MOSFET.

Be sure to "stop" your Serial connection and set both TX and RX pins to INPUT to prevent damage to the ESP8266 (or ATTiny) while powered off.