The 42-year-old manager said that he believes Barnsley deserved all three points from their season opener, at which the visiting Tykes held Forest to a 0-0 draw.
"I'm over the moon with how it went today," Hill told the club's official website.
"Did we create enough opportunities to win the game? Yes we did.
"Clean sheets do win games and I know the supporters have embraced the way that we are trying to play and the way that we are set up.
"You could tell from their reaction before, during and after the game that it was a very good performance."
Hill replaced Mark Robins as Barnsley manager in June.