Liverpool stretched their lead at the top to a massive 16 points, despite a lethargic first 45 minutes.
Fresh from a 1-0 Champions League victory over Paris Saint-Germain, Liverpool boss Arne Slot made just three changes.
However, the Dutchman was forced into making three half-time alterations to kickstart the champions-elect.
“The manager was like going for us – but that’s something you need sometimes. Because first half, the game was sloppy,” said Salah.
Southampton have taken just nine points from 28 games all season, but stunned Anfield when Will Smallbone fired the visitors in front in first-half stoppage time.
Alisson Becker was Liverpool’s hero at the Parc des Princes on Wednesday but the Brazilian and Virgil van Dijk got in each other’s way to allow Smallbone to roll into an unguarded net.
Slot turned to Harvey Elliott, Alexis MacAllister and Andy Robertson at the break, and the changes had the desired effect with two goals in three minutes before the hour mark.
Darwin Nunez levelled with a sharp near-post finish from Luis Diaz’s cross.
Nunez was then upended inside the area and Salah converted from the spot.
The Egyptian slotted home a second penalty two minutes from time to take his tally for the season to 32 goals.
Second-placed Arsenal have two games in hand on the leaders, the first of which comes away to Manchester United on Sunday.
But it appears a matter of when not if Liverpool will clinch just their second league title in 35 years.
Forest pulled four points clear of City in third thanks to Callum Hudson-Odoi’s late winner at the City Ground.
The former Chelsea winger fired past Ederson on 83 minutes from Morgan Gibbs-White’s pass to spark jubilant scenes of celebration as Forest beat the English champions for the first time since 1997.
Two-time European champions, Forest have not experienced the continent’s top competition since 1980.
But they are closing in on the Champions League with a top-five finish almost certain to be enough thanks to the strong performance of English teams in Europe this season.
“We are not here to joke”, Hudson-Odoi posted on social media. “Goal, MOTM (man-of-the-match), three points!”
City remain fourth and are looking over their shoulders after a ninth league defeat of the season.
Pep Guardiola’s men are likely to be overtaken by Chelsea, when they host struggling Leicester on Sunday.
Aston Villa closed to within two points of the top four as Ollie Watkins came back to haunt Brentford once more in a 1-0 win in west London.
Brentford goalkeeper Mark Flekken let the England striker’s effort fly through his legs on 49 minutes as Watkins struck for the sixth time in six games against his former club.
Brighton boosted their own Champions League hopes courtesy of Joao Pedro’s 98th-minute penalty to beat Fulham 2-1.
Raul Jimenez’s strike opened the scoring for Fulham, but Jean Paul van Hecke levelled before half-time.
Pedro’s spot-kick lifts the Seagulls to sixth and within one point of City.
Crystal Palace shrugged off the loss of top scorer Jean-Philippe Mateta to a serious ear injury to beat Ipswich 1-0 thanks to Ismaila Sarr’s winner.
The Eagles have won nine of their last 11 games in all competitions and climb to 11th.
Wolves edged six points clear of the relegation zone after a 1-1 draw against Everton.
Marshall Munetsi cancelled out Jack Harrison’s opener for the visitors at Molineux as Everton extended their unbeaten league run to eight games. (AFP)
