Ashley Westwood gave Sean Dyche's men the lead after just six minutes when his whipped corner flew straight into the net.
But the hosts rallied, Firmino tapping in the equaliser 13 minutes later after a defensive mix-up between James Tarkowski and goalkeeper Tom Heaton.
Sadio Mane's fine finish on 29 minutes stretched his consecutive scoring streak at Anfield to six PL matches to make it 2-1.
With 23 minutes left Firmino slotted in his second before Johan Berg Gudmondsson's stoppage-time effort gave Burnley hope.
However, Mane had the last word for Jurgen Klopp's side.
Second-placed Liverpool move on to 73 points, one behind leaders Manchester City, while Burnley lie in 17th place, two points above the relegation zone.