Happiness can't surely bring about satisfaction. One may become happy due to positive events he or she experiences like when watching comedy sitcoms. Feeling satisfied about the show is a different thing. An individual will only experience this when certain criteria are met such as the originality of the joke and the manner of its delivery.
However, it is guaranteed that satisfaction induces happiness. For example, finishing a novel or an art piece. Experiencing happiness and satisfaction is relative from one person to another.