To get to this, we may need to go back to the substantive, *Glück" (with common roots to the English luck), which has amongst its meanings both good and bad luck (so both "happyness" and a notion of "success", so basically has some notion of (positive or negative) fate (Schicksal), so the jump to "successful outcome" isn't really a big one. (Note "I had great luck growing roses in the greenhouse" wouldn't transport much about happiness, but rather final success in English as well).
Nach einem turbulenten Flug und einem langen Bustransfer sind wir nun glücklich an unsrem Urlaubsort angekommen
(typical postcard phrase) wouldn't mean "we arrived in a happy state", but rather "we finally arrived [after a bit of a struggle]". So yes, this meaning is common, and in no way regional. Note I can't imagine usage in an attributal form (as an adjective) - "glücklich" would commonly be used as an adverb in this meaning (and Duden lists it as an adverb in this meaning as well, but also lists it as colloquial).