The Reds followed up their superb 3-1 victory over Manchester City before the international break with a 12th win in 13 league games so far this season, a result that preserves an eight-point lead at the summit.
Palace looked like it had secured a deserved point when Wilfried Zaha cancelled out Sadio Mane's opener with eight minutes remaining.
Firmino had other ideas, though, prodding home after a goalmouth scramble in the 85th minute to secure all three points for Jurgen Klopp's men.
The hosts more than held their own in the early stages and could have taken the lead after 17 minutes, Jordan Ayew prodding just wide from Cheikhou Kouyate's cross.
They continued to look the most threatening throughout the first period and thought they had forged ahead three minutes before the interval.
James Tomkins's back-post header was ruled out, however, after VAR (video assistant referee) had deemed Ayew to have fouled Dejan Lovren as Luka Milivojevic's free-kick was whipped in.
Moments after shooting wide after being played through by Jordan Henderson, Mane handed Liverpool the lead.
The Senegal international's shot from 12 yards took a deflection and bobbled in off both posts after Vicente Guaita's weak wrist had failed to keep it out.
Palace responded well to that setback and forced Allisson into a string of saves.
There was little the Brazilian could do about Zaha's leveller, with the Palace talisman collecting Andros Townsend's pass and lashing into the bottom-left corner.
Liverpool bounced back in typical fashion, however, as Firmino forced home from close range to break Palace hearts.
Zaha should have restored parity again in the dying stages, but lashed wildly over after being found at the back post, much to Klopp and Liverpool's delight.