After the own goal and Matip's header, Salah got his in first-half added time, and he may not find an easier goal.
There is a feeling he is trying too hard to score at the same remarkable rate he managed last season but Klopp has no concerns and insists Salah is working harder than ever before.
If West Ham can produce an upset against Chelsea, they could also leapfrog Hughes' team.
José Mourinho: Manchester United’s improved defence not just due to defenders
United took the lead midway through the first half when Fred fired in his first goal for the club from outside the box. We have to win today because after that it will be hard to close the gap.
Southampton, meanwhile, has struggled this season, winning only one of its five league fixtures, that being a 2-0 victory over Crystal Palace on September 1. The Reds are cruising now.
The Saints could not get to the break for some respite as Shaqiri, who was surprisingly hooked at half time despite his contribution to two goals, whipped a 30-yard free kick against the crossbar, and Salah raced in to tap home the rebound from only a yard out.
In the day's late game, goals from Harry Kane, from the penalty spot, and substitute Erik Lamela earned Tottenham Hotspur a nervy 2-1 win at Brighton & Hove Albion, pushing Spurs into fifth on 12 points at the end of a trying week. Subs: Milner 6, Gomez 7, Keita 6. Subs: Armstrong 5, Bednarek 5, Austin 5.
Southampton didn't have a shot on target until the 91st minute.
In six Premier League matches, the team has allowed just two goals.
Opta point out that Liverpool are unbeaten in their last 23 Premier League games at Anfield (W15 D8), their best run in the competition since August 2009, and have kept a clean sheet in each of their last seven Premier League home games.