Moore, 28, levelled on 74 minutes with a powerful header past Yann Sommer following an effective short corner routine.
The Welsh striker was wearing a headband after a first-half head collision. He had been impactful throughout the game, having forced Sommer into an impressive one-handed save in the opening stages.
"It's great personally, we'd have liked to win but we'll take a draw to put us in a good position," Moore told BBC Sport after the match. "It's never good going a goal down but to get one back and see the game out is a big positive for us.
"Obviously this is a big occasion and I've loved every second of it."
Wales were captained by Gareth Bale, who is one of eight players in his side's 26-man squad that appeared in their fairytale semifinal run at Euro 2016.
The Swiss striker could have created a second minutes later when right-back Kevin Mbabu fired wide from his lay off.
Wales came alive shortly after, with Moore's headed strike capping off a period of sustained pressure.
Joel Ward made a couple of crucial saves late on, first tipping over a late header from Embolo and then producing a superb fingertip save from a Mario Gavranovic strike, although it was later ruled offside.
Switzerland striker Haris Seferovic thought he had netted an 85th-minute winner but VAR ruled it out for offside.
Wales next face Turkey in Baku on Wednesday, while Switzerland take on Italy in Rome hours later.