James Milner converted from the spot before half-time following a clash in the box between Mamadou Sakho and Mohamed Salah that referee Michael Oliver deemed worthy of a foul.
The Eagles were brought down to ten men in the second period when Aaron Wan-Bissaka was sent off with 15 minutes to go after he brought down Salah when the Egyptian was through on goal.
Sadio Mane wrapped up the victory late on despite a wave of Palace attacks as the Reds back-four stood firm to earn Jürgen Klopp's side back-to-back clean sheets.
Liverpool were certainly worthy winners on the night as they joined five teams including Manchester City in securing maximum points from their opening two games.
Chances for both in cagey first-half
Alisson Becker featured in his first ever Premier League away fixture and the Brazilian got a more accurate test tonight of what the gruelling nature of the English league entails.
Liverpool started firmly with their foot on the gas and Salah was found expertly early on by Mane who slid the Egyptian through but was denied by the impenetrable Sakho who came in to deny the Liverpool forward.
The Reds were required to hunt in packs whenever Eagles talisman Wilfred Zaha received the ball.
The 2017/18 Eagle of the Year looked to get a few sniffs early on providing a fearsome opponent for young England international Joe Gomez.
The move of the match in the opening 45 was orchestrated by summer signing Naby Keita who sent Jeffrey Schlupp to the cleaners with a clever drag back before unleashing Salah with a perfectly timed pass.
Salah's inch-perfect control was let down by the finish as his attempted a lob that had too much on the ball to find the Palace net.
The hosts so nearly got the lead when a poor Keita pass fell to former Liverpool striker Christian Benteke who set up Andros Townsend who cannoned a left-footed effort that completely bewildered Alisson before striking the top of the crossbar.
Andrew Robertson was a menace once again down the left-hand-side and even saw a rarity to his game when the ball fell to the Scotsman at the edge of the area before blazing his effort into Section J of the Selhurst Park main stand.
The game was certainly heating up with some flying challenges and Netherlands international Patrick Van Aanholt picked up the opening booking with a rash tackle on James Milner.
Milner penalty hands Reds the advantage
The Eagles looked to be heading into half-time with the game still at a deserved stalemate.
However, a moment of madness for defender Sakho saw some clumsy defending penalised when the French centre-back tangled with Salah before referee Oliver pointed to the spot.
The decision to award the spot-kick could be argued to come from some theatrical antics from Salah although the forward was undoubtedly clipped by the former Liverpool centre-back.
Milner took up his usual duty as the Reds spot-tacker despite Mane converting two penalties in pre-season. The Englishman was ice-cool to send Wayne Hennessey the wrong way and hand Klopp's side the advantage going in at the break.
The Reds came flying out the blocks in the second-half when Robertson let loose Salah once again who was on the back end of some Palace boos with mumbles of the word "cheat" surrounding the stadium following his theatrics earning the Reds the penalty in the first half.
Zaha continued to cause problems for the visitors drawing a foul from Trent Alexander Arnold on the edge of the area also earning the 19-year-old defender the Reds first booking of the night.
Luka Milivojević fired the free-kick goalwards and looked to be heading for the top corner, although Alisson had other ideas to force his first real save of the evening and parry away for a corner.
Another Palace set-piece looked to threaten for the hosts when Benteke showcased his supreme aerial dominance to rise and head a corner on target, but Alisson was there to deny the Eagles yet again.
Another chance fell to Salah when a half-volley fell to the forward just outside the box, he connected wonderfully with the ball that flew goalwards until Schlupp's header deflected it clear for another corner.
Benteke looked to add to his tally of seven goals against his former club when he picked up the ball from the edge of the box. However, the Belgian striker completely sliced his effort sending it painfully high and wide.
Palace restricted to 10-men
The games most significant moment of controversy came with a quarter of an hour to play and once again surrounded that man Salah.
The penalty winner looked through on goal after Firmino slotted his teammate through clean one-on-one with Hennessey. The Egyptian looked uncatchable once he was clear; however, the talented Wan-Bissaka heroicly tracked back in an attempt to deny Salah.
The young full-back dispossessed Liverpool's No.11, who was bearing down on goal, but illegally made contact with the forward and despite impressing on the night the 20-year-old talent was inevitably shown the red card.
Mané wraps up impressive Reds victory
Salah, who scored an astonishing 44 goals last season and once on the opening day, did not look at his usual full sharpness tonight - but in typical footballing fashion, the Egyptian was still involved in all of Liverpool's best moments.
The PFA and FWA Footballer of the Year set up Mane's late goal after Palace committed players forward for a corner. Salah teed the ball to Mane, who sprinted 50 yards towards goal being hunted down by Van Aanholt before going round Hennessey in ice-cool fashion to tap home into the empty net.
Liverpool were truly pushed to their limits in a fierce visit to capital in this gruelling Monday night fixture. Virgil van Dijk and goalkeeper Alisson were the outstanding players on the night for the Reds.
The Netherlands international captain was a colossus at the back, organising, dominating and leading by example whilst Brazilian No.1 Alisson made some smart saves to deny the hosts any real chances.
The win ensures the Reds mount early pressure on champions City moving up to second, whilst the Eagles first defeat of the season see's Hodgson's side fall down to tenth.
Palace were unbeaten in seven Premier League games going back to a 2-1 defeat by Liverpool on 31 March coming into the fixture and it was the Reds once again who got the better of the London outfit to make it five wins in their last six encounters against the Eagles.