They got off to a confident start against Roy Hodgson's side before dropping in intensity, allowing the hosts to pass through the press and threaten in dangerous positions.
Liverpool got off the hook just before half-time as James Tomkins' close-range header was ruled out for a push on Dejan Lovren by Jordan Ayew, who flicked the set-piece on for the defender.
The Reds started the second half strongly and took the lead through Sadio Mane, whose scuffed shot evaded Vicente Guaita and hit both posts before going into the back of the net.
Liverpool's rhythm wasn't quite there throughout the 90 minutes, understandable due to many of the squad's international commitments over the last fortnight.
They paid the price in the closing ten minutes as Wilfried Zaha created space for himself inside the box and finished across Alisson to make it 1-1.
Klopp's side have often ruthlessly responded in situations such as these this season and it was no different at Selhurst Park.
They regained the lead and secured the victory just three inutes later as Firmino netted past Guaita after the ball fell to him from close-range.
Slide through our gallery below for all the best photos from Selhurst Park.