There is a saying that "the best way to refute a gambit is to accept it", attributed to Steinitz. While sayings in chess often have exceptions, I think it does apply to this particular case. If you go by either the Stockfish evaluation (-0.8) or the results of games between highly rated players (67% win for black out of 12 games), the gambit is a mistake.
The purpose of a gambit is to trade material for piece activity and the initiative. If you accept the gambit, your plan is usually to brace for the onslaught; if you survive the attack while keeping your material advantage, you can get a winning endgame. That means you have to play solidly and be especially careful about wasting time.
In addition to 5...Nc6 as suggested by Chromatix, there are other moves which prevent this attack. For example, a simple 5...e6 also works. It helps block the diagonal leading to f7 and gives an additional flight square to your king. But more importantly, by not moving the h pawn you still keep control of the g6 square. Other moves that work are 5...Nd7 or 5...Bd7, blocking access to the queen, or 5...Qd7, advancing the queen to a protected square. None of these look particularly attractive to me but they do prevent the tactic and let you keep your material advantage.
With that said, this was a blitz game, and I think gambits work better in blitz because there is less time to think about the proper defense, and going for an endgame with an extra pawn can be tricky because you might run out of time first. So another practical way to "refute" a gambit may well be to decline it, if only because players who play gambits usually want you to accept it, and by not doing what they want they may get frustrated, and it is possible that they haven't prepared for the declined variants as well as for the accepted.