Victory against the Red Devils now means that Jurgen Klopp's side have beaten every opponent in the top flight at least once this season, to position themselves as champions-elect with four months to go.
However, Salah says that the club must remain attentive to the task at hand despite the odds being stacked in their favour and that they must not slip into complacency.
"I think we just need to keep [our] focus on each game," he told Sky Sports after the match. "We don't have to think that now we've won the Premier League or whatever, no.
"We just need to take it game by game. I think that's the only way we can win the Premier League. We struggled last season at the same time so we have to keep calm and just focus on each game.
Captain Jordan Henderson concurred, adding: "There are a lot of games left. We’ve taken it game by game for a long time now and it’s put us in good stead so there’s no need to change."
Liverpool dominated for long periods at Anfield but until Salah's strike deep in injury-time, the hosts ran the risk that they may walk away with just a point as United threatened an equaliser.
Henderson admitted that he felt his side - who saw a Roberto Firmino strike ruled out by VAR in the first half - should have put the game to bed early on, but paid additional tribute to the home fans for their support.
"The crowd were unbelievable again today but for us as players, it’s the next game and the next challenge," he added.
"The Premier League is tough, there are some tough teams, Man Utd played well at times today but overall, I felt we deserved the three points.
"1-0 is a dangerous scoreline, we know that from the Spurs game. I feel as though we should have killed the game off earlier with the chances that we had but we kept going, defended well and managed to get the second.
"It is special and we’re enjoying our football but you’ve got to stay hungry, keep learning, keep wanting more and this group of players wants to do that.
"The manager’s on us all the time to improve and we know that we can. It’s about every single day, keep going until the end of the season."