A late headed goal from Luca Antonelli sealed the three points for the Rossoblu, and Gasperini insists that his side deserved their victory.
"It was a good game in tough conditions, as there was so much water," he told Sky Sport Italia. "Inter are a physical side, but Genoa really did well and played with intensity, and this victory means a lot to us.
"Without doubt the stadium and these fans help the team and give us something more, especially when we play with this hunger.
"I think at the end of the day we were on a par with Inter throughout and had our good chances to score, as did they. As the conditions got worse, we both had to sort of throw the ball into the box and hope for the best. I believe our desire to win was decisive."
Genoa are currently 10th in the Serie A table.