Top scorer Mohamed Salah scored the victor for the Reds six minutes from time, his 29th league goal of the season.
Three minutes after the break Sadio Mane slammed home an equalizer to force Hodgson's side out of their defensive shell.
Luka Milivojevic's early penalty put Palace in front at Selhurst Park with Sadio Mane equalizing just after half-time.
The visitors were level just three minutes into the second period, however, when Mane turned a low Milner cross into the bottom corner of the net.
For the second time running Benteke should have restored the lead as Palace capitalized on a mistake from Virgil Van Dijk, the Belgian attempted a chip but was too high from goal.
Palace remain two points above the drop zone. "However, we needed to organise against the long balls from Crystal Palace".
Lallana was carried off on a stretcher at Selhurst Park and Klopp later said his player had a suspected muscular injury. To Alexander-Arnold's credit he did not let it affect his confidence and he was excellent going forward, producing a series of tremendous crosses.
BC-SOC-Crystal Palace-Liverpool, 573Salah secures Liverpool win in Klopp's 100th EPL gameAP Photo LON810, LON817, LON807, LON811, LON805, LON806Eds: APNewsNow. Substitutes: Alex Oxlade-Chamberlain (for Mane, 64) 6, Adam Lallana (for Wijnaldum, 65) 5, Dejan Lovren (for Lallana, 70) 6.
Mohamed Salah continued his sensational run of form scoring an 84th minute victor for Liverpool at Selhurst Park.
The Liverpool manager Jürgen Klopp had a similar view of the game, citing a "difficult start", but ultimately taking the view that the victory "was deserved".
Palace welcomed back Wilfried Zaha to their side and Mamadou Sakho, Martin Kelly and Christian Benteke were also in the starting line up.
Salah benefited from another assists provided by Andrew Robertson, who also helped create one of Salah four goals last time out against Watford. Salah is set to smash past that, with six games left to score the one goal he needs to take over top spot for himself.