Despite the home side dominating large periods of the match, a second-half strike from Javier Moreno secured all the points for the away side.
"We looked like we lacked confidence. After Elche's goal we looked like we could draw the game," Enrique told reporters at a press conference.
"I thought the result was a little unfair, but what counts are the goals. And in that sense, they were better than us."
Celta currently sit 14th in La Liga.