Most CMOS logic gates have input protection diodes. If your IC does, you can just use a series resistor, like a 10 or more megaohm, from battery to input. Though, the logic input needs to have no load, IE, it cannot have an internal pull-up resistor.
The battery will need to drop below 1.5v for the input to be considered a logic low unless you add a pull-down resistor at the logic input.
This will work for sensing low frequency / DC levels. It may fail if your PCB has a ton of dirty crud/really dirty flux residue all over the place.
Zeners wont work as their internal minimum current will be too high.
If you want an added voltage drop, like a zener, try using a simple old fashioned red or green led (in series with the 10 megaohm resistor (it should never light up...)). They will provide a ~1.5v(red)-2.0v(green) voltage drop with a few nano-amps whereas a typical zener will provide a 0v drop at 500na, almost like a short when drawing a few nano-amps. IE, a 10 megaohm pulldown will only be ~600na.
(Please excuse the ultra low current drop of the LEDs if I got them wrong, it's been 10 years since I used LEDs as ultra low Iq voltage drop/regulators. You can measure this by placing the LED on a battery+ and your volt meter all on it's own from the LED cathode and battery GND.)