Following Everton's win against Crystal Palace earlier today, Wolves started the game bottom of the table and came into the match on the longest winless run of any Premier League team; their last league victory being a 2-1 win against Luton Town back in April.
After a tight first half in which Wolves gave a good account of themselves, Liverpool took the lead on the cusp of half time when Ibrahima Konate rose high to head home Diogo Jota's pinpoint cross from the left.
Wolves continued to press high throughout the game though, and they got their reward for this approach in the 56th minute when Rayan Ait-Nouri capitalised on a lapse of concentration from Konate to bring Gary O'Neil's side level.
Konate attempted to see the ball out for a goal kick, but Jorgen Strand Larsen had other ideas as he harried the Frenchman and played the ball back to substitute Carlos Forbs, whose mishit shot fell to the feet of Ait-Nouri to tap home from close range.
The hosts were however only level for a matter of minutes after Nelson Semedo dragged Jota down inside the box to gift the away side a penalty, and Mohamed Salah stepped up in the 61st minute to score his fourth Premier League goal of the campaign.
O'Neil's side only mustered a few half chances following Salah's goal, and Liverpool themselves were profligate on the counter-attack as Wolves left gaps at the back in pursuit of a second equaliser.
The result sends Arne Slot's Reds to the top of the table with 15 points from their first six games, with Wolves left languishing at the foot of the table, and they are now on a run of 18 top-flight games without a clean sheet.
WOLVES VS. LIVERPOOL HIGHLIGHTS
Ibrahima Konate goal vs. Wolves (45+2nd min, Wolves 0-1 Liverpool)
The breakthrough arrives on the cusp of half time at Molineux, and it is Liverpool who take the lead through Konate.
Liverpool recycle possession after a corner kick, and the ball finds its way to Jota who works a yard of space before curling in a delicious cross for the towering French defender to rise high and head Liverpool into the lead.
Rayan Ait-Nouri goal vs. Wolves (56th min, Wolves 1-1 Liverpool)
It is sloppy defending from Liverpool and they are duly punished by Ait-Nouri who capitalises on miscommunication between Konate and Alisson to tap home from close range.
Konate tries to shepherd the ball out of play, but he is dispossessed by Strand Larsen who plays the ball back to Forbs, and after a brief scramble, Ait-Nouri is on hand to bring the home side level.
Mohamed Salah goal vs. Wolves (61st min, Wolves 1-2 Liverpool)
Parity at Molineux lasts only a matter of minutes as Jota is hauled down in the box by Semedo, and Salah converts his penalty in style to put Liverpool back into the lead.
The Egyptian missed two penalties in the league last season, but he sends Sam Johnstone the wrong way with a coolly taken strike to the goalkeeper's left.
MAN OF THE MATCH - RYAN GRAVENBERCH
The silky Dutchman has made a great start to the season under Slot, and he was once again cool and confident in the deeper midfield role today.
Gravenberch completed 95% of his passes, won three tackles and all eight of his duels, and he was always showing for the ball in tight areas; oozing class as Liverpool saw the game out.
WOLVES VS. LIVERPOOL MATCH STATS
Possession: Wolves 44%-56% Liverpool
Shots: Wolves 7-10 Liverpool
Shots on target: Wolves 3-6 Liverpool
Corners: Wolves 2-10 Liverpool
Fouls: Wolves 16-8 Liverpool
BEST STATS
WHAT NEXT?
Wolves will make the trip to London next Saturday when they face a Brentford side that are yet to taste defeat on home soil this season.
Liverpool meanwhile will be in Champions League action on Wednesday when they host Vincenzo Italiano's Bologna under the lights at Anfield, before they travel to Selhurst Park next Saturday lunchtime to play Crystal Palace.