Goals from Murillo, Nicolo Savona and Morgan Gibbs-White handed defending champions Liverpool a sixth defeat in seven league games and their heaviest home league loss in more than a decade.
Dyche has guided Forest out of the relegation zone and up to 16th in the table since replacing Ange Postecoglou in October. They have two wins, a draw and one loss in his four league games in charge.
"Of course it is to be enjoyed, it is for the fans and the players, but we have to go again. Everything's not solved but it's a good, strong marker about this group," Dyche told reporters.
"Now they are committed to the cause. You wouldn't be a Premier League player if you don't have talent but what about the hard yards? The ugly side of the game? And staying focused on what the job is - I thought that was excellent."
Forest also won 1-0 at Anfield last season, handing Liverpool their only home defeat of their title-winning season.
"Last year, this football club used it (their win at Anfield) as a catalyst and they carried on and we hope for that but it doesn't guarantee the next one," Dyche said.
Forest next host Malmo FF in the Europa League on Thursday before welcoming Brighton & Hove Albion in the Premier League on November 30.
