Rangers took a surprise lead at a raucous Ibrox as Scott Arfield rounded off an impressive move with a calm finish on 17 minutes.
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But Liverpool got level just seven minutes later as Firmino headed in Kostas Tsimikas' corner inside Allan McGregor's near post.
The match was turned on its head early in the second half as Firmino slotted home Joe Gomez's pinpoint low cross with his left foot.
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Darwin Nunez extended Liverpool's lead as he slotted the ball home on 66 minutes which came after Firmino's cheeky flick laid it off to the Uruguayan.
Jurgen Klopp then introduced Mohamed Salah to the match after the Egyptian was a surprise omission from the starting XI.
Salah, who's struggled for form this term, went ham as he fired in a stunning six-minute hat-trick - the quickest treble in Champions League history.
Harvey Elliott rounded off the victory as he fired in on the rebound with his right foot. The result confirms Rangers' exit from the Champions League.
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Salah may have got more goals but Firmino led Liverpool's fightback.
In what's been a painful season for the Reds so far, Firmino has been having a very productive one personally.
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The goals in Glasgow brought up his seventh and eighth strikes of the season.
And an interesting stat emerged after Firmino's heroics, which shows he's in very good company.
Only Erling Haaland (23) has more goal involvements for Premier League clubs in all competitions this season than Liverpool's Roberto Firmino (11 - eight goals, four assists).