Darwin Nunez added pace and penetration on Liverpool's left after missing the Crystal Palace match
Jurgen Klopp's side looked to be facing a frustrating evening at Anfield when Darwin Nunez's strike was ruled out by VAR.
However Reds captain Van Dijk made the breakthrough with a close-range header from Diogo Jota's cross and four minutes later Salah turned in a Konstantinos Tsimikas cut-back.
The victory moves Liverpool up to sixth in the table, six points behind fourth-place Tottenham Hotspur with a game in hand.
Wolves remain 15th, three points above the relegation zone, having offered minimal threat going forward.
Having failed to score in four of their previous six Premier League matches, Klopp will have been buoyed by the return of Nunez after a shoulder problem kept him out of the tame draw at Crystal Palace.
However for all the hard work of the Uruguay striker, Liverpool toiled going forward for the majority of the game before getting their breakthrough via the aftermath of a set-piece.
Trent Alexander-Arnold whipped in a free-kick that Van Dijk guided towards goal off his shoulder.
Jose Sa made a good save but Jota kept the ball in and the Netherlands defender's second header from close range found the roof of the net.
It audibly lifted the tension around Anfield and Liverpool's second goal was a much more pretty sight.
Alisson found Tsimikas with a raking pass to the left and the Greece left-back raced into the area before pulling back to Salah.
The Egypt forward guided the cross in off his thigh to move one goal short of equalling Robbie Fowler's haul of 128 Premier League goals for Liverpool.
Following the chastening 5-2 home defeat to Real Madrid and the dour draw at Palace at the weekend, this was a much-need lift for Liverpool before the game against Manchester United this weekend.
Before taking the lead, moments of quality from Liverpool at a subdued Anfield were rare.
When they did create an opening it was usually through the determination and running power of Nunez.
In the 35th minute, the former Benfica striker raced on to a pass and cut the ball back from the touchline only to see the recalled Harvey Elliott wastefully head wide from eight yards.
Just after the hour mark Nunez appeared to have put Liverpool ahead, finishing a loose ball smartly from six yards.
However, Jota was judged by VAR to have caught Nelson Semedo as he carried the ball into the box and referee Paul Tierney ruled the goal out after consulting the pitch-side monitor.
"I think you all saw the [disallowed] goal now a couple of times back and when you see it in slow motion you see the contact, but I still think it was a goal - but that's not important anymore," Klopp said.
"The players don't have a replay and for them it's a clear goal of course. Then you have to react.
"I was not worried but of course then you watch how the boys react - and the reaction was really good."
Liverpool fans may have feared this was going to be the defining moment of the game but Van Dijk intervened soon after with the goal his side's play just about warranted.
Wolves defended stoutly and with organisation but were again toothless in front of goal, hamstrung by the absence of Matheus Cunha. The Brazil striker failed a late fitness test following an ankle injury picked up at Fulham on Friday.
His replacement in the starting XI was midfielder Joao Moutinho as manager Julen Lopetegui reverted to a three-man midfield with the Portuguese veteran sitting alongside Mario Lemina and Ruben Neves.
It effectively stifled Liverpool for some time, but Wolves may well have to offer a bit more than just discipline in the coming weeks to ensure their Premier League status for 2023-24.
"It was frustrating because we didn't get anything - but we didn't deserve to get anything," Lopetegui said.
"Maybe in the first half but in the second half we didn't keep the rhythm and when they scored it was more difficult of course.
"We lost one match - it's a pity, but we have to change our mindset very fast as in three days we have another important match [at home to Tottenham]. But this wasn't a good match for us."
-- Courtesy of BBC Sport