Forget about messing around with apps. There’s a powerful voicemail handling system built into nearly every big carrier today — and that system almost always includes a system for recording a voice message and sending it directly without ringing the target phone. I will walk you through the process of doing this on Straight Talk (since that happens to be my carrier) but most carriers will have similar systems.
Great question! Although I don’t know if there is a specific function for this on your particular phone, the best way to save the message is the simplest. Turn up the sound as high as you can and record it onto a digital recorder (like you would use for a lecture), a cassette recorder or your cell phone if you have a voice recorder on it. Then save it as an MP3 on to your computer for safe keeping. You can also Google your vtech model number to see if you can find an owner’s manual. They might have a way to save a recording in there. Probably not, but it’s worth a shot.

It doesn’t matter what you save to your phone, computer or tablet — everything you keep on your devices should be considered private unless you give someone else permission to look through your files, images and apps.
-…and I’ll return your call as soon as possible. I would have answered, but I was already on the phone with Bill Gates—I swear.