Another lightning break from the Reds wrapped up three points in stoppage time, with Salah releasing Mane to round Hennessey and slot into the empty net for his sixth goal in eight league outings against the Eagles.
It's a very tough game.
Liverpool, inspired by the excellent Virgil van Dijk, weathered the second-half revival Palace had mustered until being reduced to 10 men and secured a second successive clean sheet to prevail despite offering only glimpses of their attacking best.
This was a statement win for Klopp's team, installed as City's most likely challengers, and they move into second place after building on last weekend's rout of West Ham.
Asked about the prospect of outgoing players, he said: "Nobody has told me anything [that] any transfer situation is decided".
Speaking to the press after the game, and quoted on the club's website, Klopp admitted: "It was not a brilliant football game from us", adding "Today we were not brilliant but we still won".
Bookmakers have been suitably impressed as Liverpool are now 10/3 second-favourites to win the league this season.
Palace produced an nearly instant response when Andros Townsend cut in from the right wing, racing into space towards the edge of the area and on to his left foot, before curling an effort against crossbar. "I was not 100 percent sure tonight in the first half that we would score", said the German. "OK we lost but we have to be proud".
In the end, Liverpool takes care of business and picks up a well-deserved result, and now the Reds have six points and six goals in two matches. Palace showed their top-flight experience in their opening day 2-0 win over Fulham.
Both teams debuted new signings and came boosted by great seasons a year ago, but it was an even match from the start at Valencia's Mestalla Stadium, with neither finding an edge to earn the victory. Made a save for the highlight reel on a Milivojevic free kick that was destined for the top bin.
"We can't say it was unfair, but if we had won the match it wouldn't be a insane result", Rodrigo said.
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