Liverpool fell behind after 26 minutes following a howler from goalkeeper Alisson who attempted to play a pass out from the back but directed the ball to the feet of Goncalo Guedes, who slotted it into the empty net.
The hosts, however, entered the break level after a superb cross-field Trent Alexander-Arnold pass was met by Darwin Nunez on the volley, who directed the ball past Matija Sarkic.
A defensive mix-up then handed the lead to Liverpool, with Mohamed Salah producing a calm finish after Wolves defender Toti flicked a Liverpool pass into his path.
The visitors thought they were close to knocking Liverpool out of the competition as Toti flicked the ball into the Reds' net, and sprinted away in celebration, taking off his shirt, but his effort was ruled out for offside.
The decision to disallow the goal sparked a furious reaction from Wolves manager Julen Lopetegui, who received a booking from the referee for his protests.
The boss' post match thoughts ??— Wolves (@Wolves) January 7, 2023
Liverpool's Cody Gakpo's made his debut after joining from PSV Eindhoven at the start of the January transfer window.
"He (Gakpo) fitted into the game," Klopp said. "It was not easy for a first step. We cannot expect it to be perfect, but he showed good signs. We have to get him in better positions but that will come no doubt."
"It was a good start, a dominant start," Liverpool manager Juergen Klopp told ITV. "We played really good football moments and then we concede. The goal should not happen, but we all know how often Ali (Alisson) saves our backsides.
"They had counter-attacks before then, it is always dangerous when (Adama) Traore gets the ball. We could see that. We came back with a fantastic equaliser, everything was clear at halftime what we wanted to improve and change to cause them more problems.
"Then it was a dominant performance before they got a foot in the game and we didn't win enough challenges from there. They caused us massive problems. We kept the problems in the game. The equaliser was unlucky, but he should not be able to work into our box."