The Eagles managed to hold the Premier League title challengers at arm’s length for the majority of the first-half, with Dean Henderson barely tested in the opening half-an-hour – and at the other end, Palace on several occasions came close to catching City out on the counter.
But once Antoine Semenyo and Omar Marmoush scored in quick succession, the three points did seem destined to remain in Manchester – although Palace did threaten to capitalise on some mistakes at the back in the second-half.
Lacroix told Sky Sports: “It was quite difficult. When you concede three goals against an opponent, it's always difficult.
“We showed some good stuff also, but today I think we were not at our top level, and when you play against this type of team, you have to be better than every single day.
“I think we did well in intensity, but maybe not in the right moments. We opened some space where we don't have to open space – and like I said, against big teams, you can't make mistakes.”
Lacroix was asked whether his team’s impending European Final – in two weeks’ time in Leipzig – was in any way a distraction, but rebutted: “No, I don't think so.
“I think we’re taking it game by game because if we say ‘okay, the Premier League is finished for us, we don't want to play’ – we won’t be ready for this really important game.
“We take all the games really seriously, but today you can see that we made some mistakes and they are really good, so we have to accept it.
“We also didn’t take our chances. That's also difficult.”