The centre-back recently signed for the Eagles from VfL Wolfsburg on a five-year deal and despite this being just his third appearance in red and blue, he kept a clean sheet and made more clearances than anyone throughout the game.
Palace are still awaiting their first Premier League win of the season but Lacroix says they will only get better as the season continues.
“I think we showed very good character in the second-half,” Lacroix told Canal+ after Saturday’s draw. “We could have scored, we could have finished the game. We had some very good chances.
“After that, it’s always important to win points. It’s precious – especially at home. Being strong at home is what you want, and against a good Manchester United team, I think we played a very good game.
“Personally, I’m new to the team. We have to learn to face some challenges, but I think it’s going to be better. We were much better (in the second) than in the first-half of the game – we managed to play better, and that’s why we had chances.
“We also have a very good team. We have quality, whether it's in front or behind. Everywhere, we have quality, and we’re going be dangerous. That's what we wanted to do against Leicester as well.
“We knew we might not get the ball as much (against Manchester United). But on the transition, we had to be aggressive. We should have killed the game. We didn't do it, but we're still happy to have drawn this game.”
Palace face Everton next weekend after a week off and Lacroix believes it is the perfect opportunity to kickstart the season.
“We’re having chances, we create great chances, but we’re a little behind in the last move,” he explained.
“I think it’ll be a good fight against Everton. We have to aim to go there to get the three points, and start our season properly.”