Nunez's goal ended a drought of 10 games without one, prompting fans to chant his name.
He drew a penalty three minutes later when Smallbone caught him on the back of the ankle in the box, and Salah unleashed a blistering shot from the spot past Aaron Ramsdale into the inside side netting for his first of two penalties.
"And now you're going to believe us, we're going to win the league," sang the Anfield faithful after the goal.
Liverpool's talisman completed his double with a second penalty in the 87th minute after a handball from Yukinari Sugawara.
Salah extended his league scoring lead to a remarkable 27 goals for the season, and 32 in all competitions.
The afternoon marked Liverpool's only Premier League game in March due to a busy month that features appearances in the Champions League round of 16, the League Cup final and the FA Cup quarter-finals plus the international break.
A victory looked in doubt for the first half however, with Southampton squandering an excellent chance when Mateus Fernandes swung a cross into the box and Albert Gronbaek lunged but could not quite get to the ball.
Harvey Elliott, who scored with his first touch in Liverpool's 1-0 Champions League win over Paris St Germain on Wednesday, made an immediate impact coming off the bench in the second half, forcing Ramsdale into a diving save after a corner.