Two quickfire goals by Marek Hamsik and Edinson Cavani early in the first-half were enough to help the Italian side to their first win in the competition.
Speaking after the match Garrido admitted the goals could have been avoided but is happy with his team's efforts.
"Their two goals came from mistakes and in a demanding match like this that's a deciding factor. We can be proud of the rest of the match because we fought all the time," he was quoted as saying in UEFA.com.
"The fact that we conceded two goals in three minutes can't be used to rate the team's performance. Before their first goal, we were attacking and looking for an away win.
"After their two goals, my team had to make an extra effort and get forward to look for goals, but we lacked determination in front of goal."
"Today we brushed off the fear we had in our first match and we played a good game. It's a great step forward to cast off that fear and trust in our potential.
"Our goal is not out of reach yet and we're capable of beating all of our rivals."
The defeat leaves Villarreal bottom of Group A.