The Reds were made to work hard for the win, seeing a number of attempts kept out by opposition keeper Wayne Hennessey across the 90 minutes.
Palace, winners here three seasons running prior to today, wasted their one big chance which fell Christian Benteke's way, as their wait for a first league point of 2017-18 goes on.
There was plenty of focus on both sides' defensive ranks heading into this match, having each shipped three times in last week's opening round of fixtures, but chances were at a premium in a quiet first half.
Milner opted to take on the shot from an angle when slipped through and only managed to fire straight at Hennessey, while Matip could not direct his header on target from close range.
Andrew Robertson set up that big opportunity and, on what was his competitive debut for the club, proved to be the liveliest player down the left-hand side in the first half.
Robertson's cross 37 minutes in was crying out for eventual matchwinner Mane to volley home from five yards out, but the Senegalese - under a bit of pressure from Joel Ward - failed to make the desired contact.
Liverpool had far more urgency about them as they went in search of a breakthrough early in the second half, very nearly finding one when Daniel Sturridge saw his 25-yard volley force Hennessey into a smart stop.
Hennessey was also needed at full stretch to push aside Jordan Henderson's shot from similar range, but Palace could just as easily have taken the lead as ex-Reds striker Benteke somehow powered the ball over the bar after being perfectly picked out by Ruben Loftus-Cheek.
The hosts' pressure finally told 17 minutes from time when, after a couple of shots were blocked aside in front of goal, Mane finally managed to prod the ball past Hennessey following an exchange of passes with substitute Dominic Solanke.
Mohamed Salah, a surprise sub this afternoon in a Reds side showing five changes from the off, should have sealed the points late on but could only pick out Hennessey with his back-post header.
Firmino was also thwarted by Hennessey, while Solanke and Robertson tried their luck at the death, but in the end one goal was all it took for the Reds to end their home hoodoo in this fixture.