"If you had told me five weeks ago that we would finish third, I would think it is not possible," Klopp said after the game.
"Credit to the boys, I can't believe how it worked out in the last few weeks. From outside you cannot understand it. In the hardest moments you can learn the most. We really stuck together all the time, no arguments."
Liverpool dominated possession with over 66% in the first half and enjoyed the majority of shots on goal.
However, it took Mane until 36 minutes to tap the ball in from close range following a Roberto Firmino corner.
The goal marked Mane's eighth straight goal against Palace, and his brace on Sunday took his total goals against the side to 12.
Palace sought an equaliser in the second half but none materialised as Liverpool took advantage with a second Mane goal.
Salah, who went into the day tied on 22 goals with Tottenham Hotspur striker Harry Kane in the goal-scoring charts, selflessly passed the ball off on the edge of the box to Mane, who fired it home on 74 minutes.
The result gave Palace manager Roy Hodgson a loss in his final match with the club. Hodgson, who announced he will step down after the season, took charge of Palace in September 2017 and recorded their best Premier League points total (49) in 2018-19.
There were also farewells on the Liverpool side with midfielder Georginio Wijnaldum given a strong send off from the crowd and his teammates. The Dutchman's contract at the club runs out in July and no new deal has been signed.
"A wonderful player. From the first moment it clicked. Very smart, open person. Emotional for me, I lose a friend and we'll miss him but it's normal in football. Not always nice but normal. He will find a great place, whichever club is interested," said Klopp.
Retiring kitman Graham Carter was also given a guard of honour after the final whistle and was paraded around the field by Klopp.
Meanwhile, Kane scored in Spurs' 4-2 win over Leicester City to take his scoring total to 23 and win the Golden Boot over Salah. That win over Leicester doomed the Foxes to fifth place and a Europa League spot, with Chelsea securing a fourth and final Champions League despite losing to Aston Villa.
Manchester City and Manchester United had already booked their places for the Champions League. West Ham United will join Leicester in the Europa League, and Spurs will go the Europa Conference League.