After losing nine of their last 12 games in all competitions for the first time since 1954, Liverpool stopped the rot thanks to record signing Alexander Isak's first Premier League goal and Cody Gakpo's late finish.
The victory follows a dreadful run which piled pressure on Slot, who delivered Liverpool's 20th English league title last season.
But the Dutchman also warned that he saw similar failings from his side, which contributed to their recent results, with Isak and fellow summer signing Florian Wirtz both guilty of missing good opportunities with the score at 0-0.
"There were also a few things similar to other games of ours and a few things that were different," he told reporters. "The things that were similar were that, again, we couldn't score our first chances.
"The good thing was that today, for the first time in many, many, many games, the first chance of the opponent didn't go in as well."
Slot backs Isak and unused Salah
Wirtz has endured a difficult start to life at Liverpool, but was impressive on Sunday, starting in place of unused substitute Mohamed Salah, who began a Premier League game on the bench for the first time under Slot.
Slot also gave his backing to Isak, whose disrupted pre-season due to his drawn-out move from Newcastle United has left him short of match fitness.
The focus on Isak's lack of goals is "a bit unfair to him because ... in general, a number nine plays more minutes than I've played Alex until now because of the obvious reasons", Slot said.
While Wirtz and Isak have come under fire during Liverpool's torrid spell, Salah had seemingly remained a guaranteed starter even as he visibly struggled for confidence.
Salah had not started a Premier League game on the bench since April 2024, when he and then-boss Jurgen Klopp had a touchline row in a 2-2 draw at West Ham.
But Slot played down the significance of his decision, saying: "Mo has had an unbelievable career at this club and will have a very good future at this club because he's such a special player.
"(We have) four games in 10 days with only 14 to 15 outfield players available for us, then you have to decide once in a while to make a certain line-up."