I was just playing a live chess game.  My opponent had a king and a pawn.  I just had a king.  It was his turn, but he could not prevent me from taking his pawn in the next move, which of course would have been a draw.  He resigned instead.  So, I won a game with only a king.

Is this allowed in chess rules? or, is this a  chess.com bug?

It's not a bug. Any player can resign at any time.

Interesting.  I would have expected the rule to be similar to timing out.  i.e., if he had timed out the game would have been a draw.

I guess it is possible to win with only a king...

I just found the game, this one I suppose:

http://www.chess.com/livechess/game.html?id=221170096

So he resigned before you took the pawn. If he had waited until you actually took the pawn then I wonder whether the software would automatically make it a draw. That would make sense, but as long as the pawn exists on the board there are winning chances so the program can not make it a draw. Anyway, you certainly got an easy win.

Yes.  The softrware would give an automatic draw after I took the pawn, due to insufficient material. It would also give a draw if he timed out.

I can only guess that he resigned to keep his rating down.