The Spaniard took part in just one training session following his move from Newcastle United, before being selected in the starting line-up by Reds manager Kenny Dalglish.
Hosts Liverpool drew the match 1-1 but the 25-year-old says he was happy with his start to life at Anfield.
"I am very happy because I only arrived the day before and I played my first game," said Enrique.
"I am very happy [with my debut] but I am not really happy with the result. I think in the next few games, if I continue to play, it will be easy to understand players of this quality."
Enrique reportedly cost Liverpool £8m.