Suarez was last seen in front of the home fans during his infamous incident involving Chelsea defender Branislav Ivanovic in April, but the Uruguayan forward - who paraded his new-born baby Benjamin to fans before the contest - showed relegation-threatened Palace who is the daddy with a fine contribution amid a rich return.
Suarez's appetite for a summer move away from the club seemed a distant memory as he opened the scoring on 13 minutes when he slipped but somehow threw out a boot to poke Jose Enrique's cross into the corner of the opposition net on 13 minutes.
Captain Steven Gerrard scored his first goal of the season from the penalty spot on 38 minutes after an assistant referee had deemed Dean Moxey to have pulled back Raheem Sterling. It was a harsh call for the visitors, with what was a slight contact at best - and one that looked to have been outside of the Palace area.
Liverpool's lack of intensity was obvious in the second half as Dwight Gayle headed a consolation for the visiting side on 76 minutes, but not much will worry Liverpool, who occupy their highest league position at this stage of the season since 1996.
PRE-MATCH - SUAREZ STARTS FOR LIVERPOOL: Luis Suarez starts at Anfield for the first time since he decided to chow down on Branislav Ivanovic.
13' - SUAREZ CLASS: Suarez nets for Liverpool from six yards out. The home side lead. He slipped and fell before somehow getting back on his feet to net from Enrique's cutback. Fine finish.
17' - STURRIDGE OPPORTUNISM: Sturridge twists and turns in the Palace box. Delaney all over the place as the England man bangs the shot into the corner of the net with his left foot. 14 goals in 14 games now for him.
28' - PALACE SHOW SIGNS OF RECOVERY: Kebe with a surging run into the Liverpool box, but he can't finish as Kolo Toure clears before Mignolet somehow keeps out a Kebe effort after Jerome deflected it towards goal.
33' - SAHKO SO LUCKY: Jerome with a fine cross, but Sakho almost shanks the cross into his own net in rushing the clearance. So fortunate to escape an own goal there.
37' - PENALTY TO LIVERPOOL: Sterling taken out of the action late by Moxey. Looked like it was just outside of the box. Very harsh on Palace. Was outside of the box, but the referee Anthony Taylor listens to a linesman.
38' - GERRARD COOLNESS: Gerrard plants the penalty to his left, Speroni flew the other way. First goal of the season for the Liverpool captain. Liverpool lead 3-0.
41' - MOSES MISS: Moses knocks the ball off the bar from four yards. Brilliant work by Suarez again with the cross. How did that stay out?
68' - GAYLE MISS: Glorious chance for Gayle after sloppy Liverpool play, but he fires the effort high of the mark. Poor finish really. Liverpool toiling for any intensity.
77' - GLORIOUS GAYLE: Campana's free-kick is glanced into the net by Gayle's head. No chance for Mignolet. Palace rewarded for a spirited performance in the second half. They trail 3-1.
TALKING POINT: Just how good are Liverpool? Another week, another win. Far too early to suggest Brendan Rodgers's side are genuine title contenders, but their front pair is one of the best in Europe. If this particular duo stay fit, who knows where the weeks ahead might take the home side? One thing is for sure: they're a lot of fun to watch at the moment.
Luis Suarez and Daniel Sturridge started a Premier League match together for the ninth time. The SAS have now scored 11 goals combined in those games - Suarez six goals and Sturridge five goals.
Crystal Palace lose six of their opening seven Premier League matches for the first time in their history.
Liverpool are the highest first half scorers this season with 10 Premier League goals and joint best defensive side in the first half after conceding just once.
Liverpool are the lowest second half scorers in the Premier League with just one goal.
Liverpool at this stage last season only had six points. This season they have 16.
Steven Gerrard's 99th Premier League goal means that he has now scored in 15 consecutive seasons for the Reds
MAN OF THE MATCH
Jordan Henderson (Liverpool): Could have gone for Suarez or Sturridge, but we'll give it to Jordan Henderson for contributing to Liverpool's dominance in the first half that provided them with the platform for a well-deserved three points.
Liverpool: Mignolet 7, Toure 7, Skrtel 7, Sakho 6, Henderson 8, Enrique 7, Gerrard 8, Sterling 7, Moses 6, Sturridge 8, Suarez 8..subs: Agger 6, Aspas 5, Alberto 6
Crystal Palace: Speroni 8, Ward 6, Mariappa 6, Delaney 5, Moxey 5, Jedinak 7, O'Keefe 5, Puncheon 5, Kebe 6, Jerome 6, Chamakh 5.subs: Campana 6, Bolasie 7, Gayle 7
MATCH IN A TWEET