Seventeen-year-old Rio Ngumoha struck a brilliant opener followed quickly by a goal from Mohamed Salah that kept Liverpool in fifth place and Slot is hoping that has sparked some momentum heading into Tuesday's second leg of their Champions League quarter-final at home to Paris Saint-Germain.
Slot's side had not won a league game since late February and were coming off humbling defeats, including a 4-0 rout by Manchester City in the FA Cup last eight and a 2-0 loss at PSG.
While acknowledging the mood around the squad had improved, Slot struck a note of caution.
"It's a better place than it was two or three days ago or a week ago, but it's also not that after we win everything is good again," he said.
The Dutchman praised the supporters for the raucous atmosphere on Saturday and is asking for more of the same when Liverpool take on PSG at Anfield next week.
"This was a massive win not only for the league but looking forward to Tuesday, not only for the players but also for the fans. After a 4-0 loss and a 2-0 loss everybody could do with this win," Slot told the BBC.
"One thing is clear is that on Tuesday we need our fans. There were fans in Paris but we had a very hard time.
"Anfield has shown many times that it can lift the team to another level, and we need that again. Half an hour before the game the players need that support and need to feel it is going to be another special evening."
The 17-year-old Ngumoha became the youngest ever Liverpool player to score in the Premier League at Anfield and had fans on social media wondering if the teenager might start against PSG.
Slot praised Ngumoha's historic strike, highlighting the teenager's growing influence.
"One of his strengths is dominating the one-v-one," Slot said. "We've seen this season as he gets more and more playing time ... that's what he deserves.
"When he plays, he plays really well and today he scored, which is vital in football. Goals matter a lot."
Salah scoring in his first game at Anfield since announcing he would leave the club at season's end after nine glittering years struck a chord with the supporters, who chanted his name.
"It is important for Mo that he scored this goal, but it was mainly important for the team and the club that we won this game," Slot said.
"For Mo to score a finish that he has done so many times, inside foot, far corner ... I'm happy he scored a vital goal for himself and the team."
