The Spaniard, who was linked with a move away from Anfield after losing the left-back position to James Milner, has started in two out of three of the Reds' Premier League matches this season and featured against Hoffenheim in the qualifying round of the Champions League.
Moreno is expected to share the left-back slot with Andy Robertson on a rotational basis going forward, as Klopp has moved Milner to midfield.
"When we signed Andy Robertson it gave me more strength and made me focus even more," the Spaniard told The Telegraph.
"In the summer training sessions the manager started using Milner in central midfield, so that was also encouraging. I knew there was an impetus to keep working and things would work out.
"I felt good and thought I can stay here and I can do a job here. I knew in pre-season I had a future here. Obviously I feel much happier now."
Liverpool, who sit in second place in the Premier League table, take on Manchester City this weekend.