Dan Ndoye put Nottingham Forest ahead in the 24th minute before Alexander Isak levelled on the hour with a tap-in from Florian Wirtz's low cross.
Morgan Gibbs-White restored Forest's lead from the penalty spot after Alisson fouled Neco Williams, but Munoz, handed his full debut, thumped home an 82nd-minute equaliser to rescue a point for Andoni Iraola's side.
Alisson was booked for protesting the penalty award and later kicked a hole in one of the substitutes' boards at full time, having also been frustrated by delays getting a triple change ready before the incident.
Iraola felt his side's slow start cost them.
"It's not the result we want. I think two very different halves," he said. "We were poor in the first half, even being poor, we were unlucky in terms of their goal. We were much better in the second half.
"The intensity was different - if you are going to lose, you have to lose like in the second half. A lot more intense, more threatening, happy with the second half, but it was late in the game."
He also addressed the delay before Forest's penalty: "I will analyse the penalty, yes, but a professional fourth official cannot take three minutes for the subs. I was angry because in the previous stop we couldn't make it. And before the throw-in I said 'let's do it now', but I don't know how to explain it - it has taken ages to prepare two boards."
Nottingham Forest boss Oliver Glasner was satisfied with his side's display despite failing to hold on for a first win of the season.
"I think we all have the same feeling, proud of the performance, proud of how many chances we created and proud of the reaction after we conceded the equaliser," he said. "On the other side, we're all a little bit disappointed we couldn't get the win. I think this shows how good we performed, how well we played. Overall, a very good performance."
Glasner pinpointed a late dip as the difference.
"I could see our tactical shape work well in many parts but after 60-65 minutes, the level drops a little bit, which shows everything is not 100 per cent settled - it can't be after six or seven weeks together. We conceded the same goal more or less in the same area. We opened the middle and we know Liverpool always tries to pass into these gaps."
He added: "What I loved most was how we played forward and created the chances, especially in the first half. I think we should have led by more than one goal. But again, it's still the beginning of the season, the beginning of the journey, and this performance will give us confidence."
Asked about incoming transfers before Tuesday's deadline, Glasner said: "I am hopeful of new signings. The club are working hard behind closed doors, I'm confident we will have two signings. We have Liam in and hopefully one more."