Let us say we have the following conditions with our op-amp:

Vcc = 0
Vee = 0
Vin = 1ms 5V pulse
Internal ESD protection diodes to each rail

Since my op-amp is unpowered, then the input signal is forward biasing the ESD diode from input to the positive power supply rail. Depending on the current from the signal source and perhaps the sinking ability of the positive power supply, wouldn't Vcc of the op-amp be connected to the Vin signal minus a diode drop thus providing power for a brief moment?

 

I've seen some parasitic transistor circuits, so something inside the op amp might respond. The number of transistors makes it unlikely that you would see any output. Too many places needing current sources and so on.

Got an o'scope? It would be easy to set up an experiment.

 

It's not uncommon that components can be "backpowered" through the protection diodes. It can also happen with logic chips, especially those with FET outputs that inherently contain body diodes.