Liverpool, holders of the FA Cup and debuting new Dutch forward Cody Gakpo, started brightly before a number of errors at the back were eventually punished when Alisson's pass was easily cut-out by Goncalo Guedes, who enjoyed an easy finish to put Wolves into the lead.
The visitors looked bright with momentum, but they couldn't hold out before the break as Darwin Nunez expertly guided the ball past Matija Sarkic in added time after an excellent ball from Trent Alexander-Arnold, restoring parity just moments before the half-time whistle.
Liverpool didn't take long to get themselves into the lead after the restart as an error from Toti Gomes was punished by Mohamed Salah, and a moment of redemption followed as Alisson saved from Rayan Ait-Nouri.
Wolves got back on level terms through Hwang Hee-Chan after some naive defending by the Reds, who then had to get past a VAR review when Toti Gomes saw what could have been the winning goal ruled out for a marginal offside.
The two sides will play once more at Molineux to see who advances to the fourth round of the competition.
A positive debut for Gakpo indicates that Liverpool have finally found a strong alternative on the left-flank, as the Reds continue to deal with the injury absences of Diogo Jota and Luis Diaz. Ibrahima Konate also impressed ahead of a spell where he is set to feature regularly with Virgil van Dijk sidelined. A goal for Nunez could also be the catalyst for a promising run of form.
Too many careless moments saw Liverpool questionable with their decision-making, not least for the opener where Alisson looked to still be in the holiday season, gifting a goal to Guedes. Those errors will need to be avoided if the Reds are to retain the FA Cup this season ahead of the replay at Molineux.
Manager rating out of 10
Jurgen Klopp, 7 -- The Liverpool manager showed his intent to go far in the tournament by deploying his strongest possible side, and Nunez looked to benefit from the addition of Gakpo on the left-flank. Positive substitutions also gave his side the best chance of getting a result from the game.
Player ratings (1-10; 10 = best, players introduced after 70 minutes get no rating)
GK Alisson, 5 -- A moment of madness from Alisson saw his pass easily picked off by Goncalo Guedes, who finished to put Wolves ahead. Produced a strong stop against Rayan Ait-Nouri to keep Liverpool in the lead in the second half, but he was unfortunate with the second goal that the Reds conceded.
DF Andy Robertson, 6 -- A tough first half for the Scotland international saw him battle against Adama Traore, who enjoyed some success down his flank in the opening 45, but the left-back was able to swing the pendulum back into his favour after an assured second-half display.
DF Ibrahima Konate, 7 -- Maintained composure when isolated and won the majority of his battles. Konate was back to better form after a poor display against Brentford last time around. Liverpool's best defender on the night.
DF Joel Matip, 5 -- Almost caught out by Guedes when trying to dribble past him from inside his own box. That exemplified some poor decision-making at times from a normally consistent defender.
DF Trent Alexander-Arnold 8 -- Alexander-Arnold got forward as much as possible and enabled space for Henderson and Salah. His assist for Nunez was a moment of magic to help secure the equaliser, and he also defended well when called upon.
MF Fabinho 6, -- The Brazil international took up intelligent positions as he found a positive balance between pressing aggressively from the number six role without giving up space centrally on the counter-attack.
MF Thiago 6, -- Careless play before the opener saw Thiago concede possession under no pressure from a poor pass, with the 31-year-old wasn't quite at his best on the night, though he did grow into the game.
MF Jordan Henderson, 6 -- The Liverpool captain got forward and picked up good positions to link play with Salah, but his influence faded into the game as time went on. Eventually replaced by Naby Keita.
FW Cody Gakpo, 6 -- The debutant provided glimpses of his quality throughout the game and contributed to Liverpool's second goal when his intended pass for Salah was helped on by Toti Gomes.
FW Darwin Nunez, 7 -- Nunez troubled wolves with his runs in-behind and produced a world class finish to ease the ball past the Wolves goalkeeper. Quiet in the second half.
FW Mohamed Salah, 6 -- Salah came in and out of the game in spells while remaining a danger, and he made no mistake with the finish when presented with a chance in the 52nd minute, but his play in the final third could have been better overall.
Naby Keita (Henderson, 68"), 7 -- Keita looked calm in possession and progressed the ball forwards well. Another positive display from the Guinea international who is still garnering match sharpness.
Alex Oxlade-Chamberlain (Gakpo, 84"), N/R -- Deployed on the left-flank in place of Gakpo.
Joe Gomez (Alexander-Arnold, 85"), N/R -- Brought on for Alexander-Arnold who had worked hard throughout the game.