Palace were reduced to 10 men when Aaron Wan-Bissaka was shown a red card in the 75th minute for clattering into the back of Salah, who was through on goal, and he exited the pitch to a standing ovation from the Palace fans.
Salah was accused of diving by Palace when he tumbled in the penalty area late in the first half at Selhurst Park.
While they could have won by more than the final 2-0 scoreline suggested, the game was tight, although Liverpool were clearly the better side and just didn't make the most of their opportunities to get the second goal that would have calmed a few nerves until Sadio Mane's late clincher.
The Reds arrived at Selhurst Park prepared for a hard night, with Klopp praising Roy Hodgson's "well-organised" side in his pre-match press conference.
When asked if he'd like to see it be brought in, he said: "I don't want VAR in this league because, in my opinion, this country has the best referees in the world".
Referee Michael Oliver's decision to award the penalty seemed generous and Hodgson was strong in his condemnation. There is no point discussing it. Sakho is a defender and he has to try and defend, which I think he did well.
Having also netted twice in the 4-0 opening-day win over West Ham United, Mane is now joint-top scorer in the Premier League, tied on three goals with Manchester City's Sergio Aguero and Everton's Richarlison. Former Liverpool boss Hodgson is likely to keep with the same side on Monday and will be buoyed by the vocal home support.