Mohamed Salah looked to have netted a winner on his Liverpool debut, but Miguel Britos headed home from point-blank range in the dying moments to secure a share of the spoils.
Liverpool -- without Philippe Coutinho, who was ruled out with a back injury amid intense speculation over his future -- fell behind on eight minutes, with Stefano Okaka taking advantage of some poor defending from a corner to head past Simon Mignolet.
The visitors hit back on 29 minutes with a superb goal from Sadio Mane, but Liverpool's defensive weaknesses were exposed again almost instantly as Abdoulaye Doucoure slotted in for 2-1.
Liverpool turned it around early in the second half, with Roberto Firmino equalising from the penalty spot after Salah was brought down in the area by goalkeeper Heurelho Gomes on 55 minutes.
Two minutes later, Salah was able to nudge home at the far post following Firmino's lofted pass over Gomes as Jurgen Klopp's men took the lead for the first time.
That appeared to have set Liverpool on course for victory, but Britos struck late to leave the Reds frustrated.