The start is like an early Twilight Zone episode

I think you mean "Where Is Everybody?", the first TZ episode and one of the best. It starred Earl Holliman, who "wakes up" (not exactly, but no spoilers in case someone hasn't seen this 55-year-old TV show LOL) to an empty town. That's where the resemblance ends, but it's such a creepy effect it could well have stuck in Serling's mind.