You are exactly right about the connection between the two sentences,
but the terminology isn't right because there's no such thing as a
'prepositional clause'. Instead, we say that the prepositional
phrase in 1 functions as an adverb because it answers the question
"Why?"
-- Why are you happy?
-- We won the game. I am happy about winning the game.
The clause in 2 that means the same thing as the prepositional phrase
also functions as an adverb which answers the question "Why?"
That's why it's an adverbial clause. It gives the reason for the idea in the main clause (
I am happy).
CJ