'This is a point and a draw, maybe we deserve a little more. In the end, though, they were punished by Lacazette's fine 82nd-minute finish.
Jurgen Klopp disagreed with the decision to disallow Sadio Mane's goal for Liverpool against Arsenal on Saturday evening but said he can not complain about leaving the Emirates Stadium with a point in hand.
But Milner's goal stemmed from one of only a handful of chances Liverpool mustered in the second half.
He's right. Improvement is nearly certainly still to come from this side, though you could say the fact that Liverpool are disappointed with a point away at Arsenal is indicative of the progress Klopp's team has made. The most joy Liverpool had going forward was when they went direct and got the ball up to Roberto Fimrino, Mohamed Salah and Sadio Mane as quickly as possible.
They were unlucky not to win the game as Palace forced mistakes in the Arsenal backline.
Henrikh Mkhitaryan headed wide, Alexandre Lacazette had an effort rightly ruled out and then flashed across the face of goal as the Gunners searched for a first-half score they've found so hard to come by this season.
What was most remarkably about this Arsenal club, though, was how many individual players stepped up and asserted themselves on the match. They remain the kings of the second half, with 19 of their 25 league goals coming after the break so far, and 10 of their last 11.
We expect Hector Bellerin to overcome his injury troubles and make a start.
"The reaction of the team was ideal", goalkeeper Bernd Leno told Arsenal's official website afterwards. The former German worldwide has averaged nearly two key passes per game this season and his ability to carve open the opposing defence will be paramount in Arsenal's attempts to break down a Liverpool defence that has conceded only four goals this campaign.
It is still early days of the new era under Emery, with Arsenal just 16 games in, but there is an air of positivity around the Emirates again, and that is success in itself.
Emery's decision to take off top scorer Pierre-Emerick Aubameyang for Iwobi had raised eyebrows, but this wasn't the first time his mid-game change has made an impact.
That also underlines the attacking depth the manager has at his disposal.
Too many times in Arsene Wenger's last few seasons Arsenal completely lost the plot against their rivals. Those huge gaps aren't closed over the course of one summer.
Yes, there was a wobble after James Milner put Liverpool ahead but there was no sign of the capitulation we may have witnessed in the past.
Emery's energy transmits itself to Arsenal's players and supporters.