Salah's goal against the Saints was his 22nd of the season, one behind top scorer Harry Kane, and the 25-year-old is hopeful he can continue to fire the Reds towards Europe.
"It was important for us to get a good start," he told BBC Sport . "When you score 10, 15 minutes in, it makes the game easier, less pressure. I’m happy about the start, but we have to keep looking forward."
When asked about his goalscoring exploits, he added: "I said in the beginning, I always try to help the team. When you score of course you help the team. So I’m not surprised, but we need to forget about the past and keep looking forward.
"We have more league games and the Champions League."
Klopp jokingly described his star attacker as "greedy" and stressed that all of his attack-minded players deserve recognition for a performance that propelled them four points clear of fifth-placed Chelsea, who have a game in hand against West Brom to play on Monday.
"I don’t want to forget Sadio [Mane] because he was involved in everything. Mo [Salah] is very good, really greedy," he told BBC Sport.
"Roberto Firmino was not bothered too much. Mo had assist and a goal, both of them did and it was deserved. Important players like that make a difference. In the second half [Georginio] Wijnaldum was spot on, Emre Can is a good character. They're all good characters.
"We don’t have a normal weekend any more. We want to finish in the top four, we need points for sure. [Arsene] Wenger said after Tottenham it was not only derby defeat but more.
"Nine points is not enough [to be clear of] Arsenal. Tottenham are strong, United strong, Chelsea strong, City champions. So we need to continue like this."