Anything with alcohol in it can make someone drunk. However, if the alcohol is burned off like in a rum cake, then that can’t make you drunk. The alcohol has been burned off and is gone away. Usually, one cocktail doesn’t make someone drunk. Although, it could make someone tipsy. The more cocktails that you drink, the more likely you will get drunk. Certain cocktails have more alcohol in them and they will make people more drunk.
Also, it depends upon the bartender. Some bartenders put more alcohol in a cocktail than what is required by its recipe. Then when the drink is served to the person, then the drink will make the person more drunk when they drink it.