The Spaniard was at fault for Manchester City's opener in the 3-1 defeat last week but he responded with a brilliant 60-yard run to score the Reds' third in Sunday's victory at Tottenham.
He considers that to be the initial down payment towards him clearing his debt from the loss at the Etihad Stadium.
"After making my debut and losing against City I had a point to make," he said.
"When I stole the ball from (Andros) Townsend and saw a way through, all I could think about was getting to their goal and scoring. I didn't care how many metres I had to run.
"I hope it isn't my only one as I hope to dedicate many more to Liverpool's fans.
"If they didn't know me before, they will from now on thanks to my goal against Tottenham."
Moreno won the Europa League with Sevilla last season before moving to Anfield in a protracted £12million transfer but he is looking forward to making the step up with Liverpool due to reacquaint themselves with the Champions League this month after a four-year absence.
"Playing in the Champions League is going to be amazing," he told liverpoolfc.com.
""It's all new for me and I'm really looking forward to it.
"Plus, we have Real Madrid at Anfield and I'll be wearing Liverpool's colours. It's going to be a unique experience.
"I saw the trophy at Melwood and I got goose bumps. Hopefully we can win it this year. Hopefully that day will come."