The Eagles were perhaps punished for some profligate finishing on Merseyside, twice hitting the woodwork and numerous times being thwarted by Liverpool ‘keeper Freddie Woodman.
In total, Palace generated 2.32xG to Liverpool’s 0.91, indicating a generally impressive performance from Palace away at the incumbent Premier League champions.
Now, Lacroix and his teammates will turn their attentions swiftly to Thursday night’s UEFA Conference League semi-final first leg away at Shakhtar Donetsk.
“I think we deserved something better,” Lacroix said of Saturday’s game. “We had the best chances, but we weren’t really efficient in front of goal.
“I think we played well. It's not easy to play like this in Anfield. We could score, I think, two goals. It’s the finish, the last step [we missed]. We have to say congratulations to the goalkeeper. He made some really good saves.
“Maybe we need more accuracy in front of the goal, but we are sure that next game we will score the goals.
“I think this match prepares us well for the Conference League. We showed a great performance. We're ready for the next match in Europe.”