Mo Salah was fouled by former Reds defender Mamadou Sakho inside the penalty area with James Milner converting from the spot just before half-time.
Sadio Mané made sure of the points late on with a goal in stoppage time.
Such is Liverpool's strength in depth, new signing Fabinho and full-back Nathaniel Clyne were not included even on the bench.
On a night when the Reds had to battle to get over the line, here is how Jurgen Klopp's men rated.
Goalkeeper and Defence
Alisson (8): Made a number of good saves, looked confident when catching and his distribution is at least as good as Ederson's at Manchester City.
Trent Alexander-Arnold (7): Coped well with Crystal Palace's wide threats - culminating in a battle with Wilfried Zaha, which he just about won - but his crossing was slightly off once again, particularly in the first half.
Joe Gomez (8): A dominant display first against Christian Benteke and then against the timid Alexander Sorloth; some excellent distribution, too, and will definitely retain his place ahead of Joel Matip.
Virgil van Dijk (9): Also dominant, and he talked Gomez through the game; oozes class whenever he puts on the Red shirt, and is such a clear upgrade on Sakho, who was typically haphazard at the other end.
Andy Robertson (7): Had a tough time in the first half when his passes and crosses went astray, but improved after the interval; managed to get himself into some good positions and did well to slow Zaha down at times.
Midfield and Attack
Gini Wijnaldum (7): Any questions about him not being strong enough away at a tough ground like Selhurst Park were quickly squashed; shielded the defence very well and will play a number of games in that number six position this season.
James Milner (7): All over the pitch as usual and held his nerve for the crucial penalty kick that opened the scoring; another who should retain his place on the pitch next weekend.
Naby Keita (8): One poor pass led to Andros Townsend hitting the crossbar from distance, but just a minute before he showed why Liverpool have bought him, spinning away from a pressing attacker before sending an inch-perfect through-ball into Salah; he is set to be a star and will only get better.
Mo Salah (7): A quiet day for him all things considered, but he was involved in the penalty being won, brought down by Sakho as he was about to let fly, and also drew a foul from Aaron Wan-Bissaka when the young full-back was sent off; will have much better days but can be content with his contribution.
Roberto Firmino (7): Still not at his best after returning from the summer break, but another 90 minutes under his belt will do him a world of good; showed some good touches but his decision making is not quite up to speed yet, although it will be ina couple of weeks
Sadio Mané (7): Brilliant in flashes but frustrating at times too; got the goal that sealed the victory which gives him a very decent record of three goals in two league matches this season.
Jordan Henderson (7): Slotted straight into the team when he came on, which is not easy when the tempo is high; if anyone comes into the team for Brighton at the weekend, it will be the Liverpool captain, who looks hungry and full of confidence.
Adam Lallana (6): Gave away a silly foul late on; kept the ball nicely during the short time he was on the field.
Daniel Sturridge (n/a)