Mohamed Salah gave the Reds the lead following a spell of pressure, scoring after a fine move.
Moments later young debutant Rhys Williams was caught out in the offside line to allow Son Heung-min in behind to finish well and level the scores.
With just moments left in the match an Andrew Robertson corner was met with a bullet header from Firmino to grab all three points; the Brazilian was over the moon as he headed straight for the supporters in the Kop.
Liverpool are three points clear at the summit of the Premier League and unbeaten in 66 home matches in the competition.
This is what Jurgen Klopp had to say in his post-match press conference, with a full transcript available from liverpoolfc.com.
- The result
''I really think we deserved the three points. We were in charge.''
- The goals
''First goal we forced a bit, second one from a set-piece was really nice as we don't have time to work on these. Really nice.''
- His side
''We don't struggle with confidence. We are a good team but we don't need to make a story of it. A performance like this on the pitch is pretty special. We need to be ready again for Palace.''
- On Firmino
''I hope he never doubts himself. So much talk about it. He scores goals and they are important ones for us. It doesn't matter who scores them. How we get there is about passing and crossing, finishing, but also making runs people don't see. Really happy.''
- On the significance
''Felt a bit like last season, I don't think about it but I can see why people say that. One of the best performances for a while. A top game, we played football from time to time. Massive three points.''