Arsenal were well and truly outclassed at Anfield despite taking an early lead through Ainsley Maitland-Niles, as the Reds moved nine points clear at the top of the Premier League table.
Firmino hit back with two quickfire goals, before Sadio Mane and Salah – from the penalty spot – had Liverpool 4-1 by half-time.
The Reds were given another penalty in the second half when Sead Kolasinac appeared to nudge Dejan Lovren to the ground, but Salah allowed Firmino to take it and he duly completed his first Premier League treble.
And Klopp singled that moment out at full-time, telling BBC Sport: "We knew in the second half that the game wasn't finished, so we wanted to control it.
"Mo Salah gave the penalty to Bobby [Firmino] and I almost cried because we all know how much Mo wants to score goals. It was really nice.
"Christmas is over but not between the boys!"
Klopp was also keen to laud Liverpool's response to initially falling behind, as they bounced back emphatically and swiftly.
"The response [to going behind] was world class," Klopp said. "We stayed calm and really forced the mistakes.
"Firmino's second when Sadio [Mane] won the ball - wow. It was perfect defending."
Up next for Liverpool in the Premier League is a trip to defending champions Manchester City on January 3.
Liverpool know that they will kick-off at the Etihad Stadium at least seven points ahead of Pep Guardiola's side, and if they can avoid defeat they will be in pole positon to claim the title for the first time since 1990.